Entanglement and Non-Locality in Games and Graphs

Arthur Mehta

Abstract

This thesis is primarily based on two collaborative works written by the author and severalcoauthors. These works, presented in Chapters 4 and 5, are on the topics of quantum graphs, andself-testing via non-local games respectively.

Quantum graph theory, also known as non-commutative graph theory, is an operator space gen-eralization of graph theory. The independence number, and Lovász theta function were generalizedto this setting by Duan, Severini, and Winter [DSW13] and two different version of the chromaticnumber were introduced by Stahlke [Sta16] and Paulsen [HPP16]. In Chapter 4, we introducetwo new generalizations of the chromatic number to non-commutative graphs and provide anupper bound on the parameter of Stahlke. We provide a generalization of the graph complementand show the chromatic number of the orthogonal complement of a non-commutative graph isbounded below by its theta number. We also provide a generalization of both Sabidussi’s Theoremand Hedetniemi’s conjecture to non-commutative graphs.

The study of non-local games considers scenarios in which separated players collaborate toprovide satisfying responses to questions given by a referee. The condition of separating playersmakes non-local games an excellent setting to gain insight into quantum phenomena such asentanglement and non-locality. Non-local games can also provide protocols known as self-tests.Self-testing allows an experimenter to interact classically with a black box quantum system andcertify that a specific entangled state was present and a specific set of measurements were performed.The most studied self-test is the CHSH game which certifies the presence of a single EPR entangledstate and the use of anti-commuting Pauli measurements. In Chapter 5, we introduce an algebraicgeneralization of CHSH and obtain a self-test for non-Pauli operators resolving an open questionposed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other thanthe maximally entangled state, and hence resolves the open question posed by Cleve and Mittal(ICALP 2012).

The results of Chapter 5 make use of sums of squares techniques in the settings of group ringsand ∗-algebras. In Chapter 3, we review these techniques and discuss how they relate to the studyof non-local games. We also provide a weak sum of squares property for the ring of integers Zn. Inparticular we show that if b ∈ C[Zn] is hermitian and positive under all unitary representationsthen it must be expressible as a sum of hermitian squares.

Chapter 1

Introduction

Quantum information theory (QIT) is a study that examines the nature of information processingthat is governed by quantum mechanics. As a field of study, it sits at the cross roads of manydifferent scientific specializations including theoretical physics, theoretical computer science andpure mathematics. Each of these specializations provides unique viewpoints and can differ fromone another in terms of notational conventions, style, and philosophical outlook. These differencesmake collaboration between experts both challenging and profitable to the study of QIT.

The goal of this thesis is to highlight collaboration across different specializations by using theirdiffering motives and techniques to ask new questions and obtain new results. These results arelargely based on two collaborative research projects, written by the author and various coauthors,on the topics of quantum graphs and non-local games [KM19], [CMMN20]. Each of these projects hasbeen accepted for publication in reputable academic journals and they are presented in this form inChapters 4 and 5. In Chapter 1, we give a high level introduction to self-testing through non-localgames and the topic of quantum graphs. We discuss how each of these works connects with the goalof collaboration across different disciplines.

Providing the correct mathematical framework is one of the most important contributionsmathematics can provide to the interdisciplinary study of QIT. In the absence of a completemathematical theory related results may appear disconnected and techniques can come across asad hoc instead of generalizable. Chapters 2 and Chapter 3 highlight this contribution. Chapter 2is not original work but serves to provide a reference for an often overlooked discussion on theoperator theory foundations of QIT. In Chapter 3, we visit the topic of sums of squares in the contextof group algebras. Chapter 3 contains some new results and this chapter concludes by connectingthe topic of sums of squares to techniques used in Chapter 5.

1.1 Non-locality and Self-testing

In May of 1935, Albert Einstein, along with his two postdoctoral students Boris Podolsky andNathan Rosen (EPR), published their now famous paper Can Quantum Mechanical Description ofPhysical Reality Be Considered Complete? [EPR35]. This paper planted the seed for one of the mostintense scientific debates of the 20-th century. Einstein, Podolsky and Rosen were motivated by thepossible implications of the Heisenberg uncertainty principle. This principle claims that there arepairs of incompatible measurements of a state that cannot both be known with certainty. Mathe-matically, this principle was derived from an operator theory description of quantum mechanicswhich represents measurements as operators on a Hilbert space. This was a severe divergencefrom classical mechanics which presupposes that there must be an underlying “reality” of any

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observable quantity of a given state. Consequently, Einstein, Podolsky and Rosen argue that theprevailing operator theory formalism of quantum mechanics was an incomplete description ofreality. In an effort to show this description was incomplete they propose the following thoughtexperiment.

Two scientists, Alice and Bob, are spatially separated and each possess one of a pair of entangledparticles. Alice and Bob can each make one of two possible incompatible measurements, call thesemeasurements M|0〉,|1〉 and M|+〉,|−〉. A global state, |EPR〉, is used to jointly describe separatedsystems of Alice and Bob. If Alice first applies measurement M|0〉,|1〉 and observes outcomei ∈ 0, 1 then Bob’s state would have collapsed so that his measurement outcome with respect toM|0〉,|1〉 would also be i with probability 1. Similarly, if Alice first applies the second measurementand observes outcome j ∈ +,− then Bob’s state would also measure as outcome j with certainty.Einstein et al. go on to argue that this shows that Bob’s particle must have the value for each ofthe two measurements predetermined and the current quantum mechanical description is thusincomplete. A central assumption in this reasoning is the idea of locality, which purports that ifAlice and Bob are spatially separated then measurements conducted by Alice can not affect theunderlying “reality” of Bob’s particle.

Is it possible to replace the prevailing operator theory formalism of quantum mechanics with amore complete model that is compatible with the assumption of locality? This was the challengeimplicit in the work of Einstein, Podolsky and Rosen. At a high level, this question is concernedwith the appropriate mathematical framework to describe quantum mechanics. The possibility of aframework for quantum mechanics that is also consistent with classical physics was eventuallyrefuted by Bell in his landmark work On the Einstein-Podolsky-Rosen paradox [Bel64]. Bell’s theoremwas later simplified using non-local games [CHSH69].

1.1.1 Non-Local Games

In a non-local game two cooperating players, Alice and Bob, interact with a third party knownas the verifier. The game consists of two sets of questions, one for each player, and two sets ofpossible answers, again one for each player. The rules of the game simply consists of an assignmentof win or lose to each possible 4-tuple of questions and answers for each player. Prior to the start ofthe game the rules are known to all players and Alice and Bob are given an opportunity to agreeupon a strategy. Once the game is to begin, the players are spatially separated so as to prohibitany communication between Alice and Bob. This condition of separation can be viewed as anon-locality condition. The verifier then randomly selects and provides one question to each playerand receives one response from each player. The players are then determined to win or lose basedupon the rules. For a more formal description of non-local games please see Section 5.2.2.

In their celebrated paper [CHSH69] Clauser, Horne, Shimony, and Holt show that the scenariodescribed above is a compelling setting to probe the limitations of locality and the implications ofquantum entanglement. They consider the following game, widely known as the CHSH game. Thequestion and answer set for each player is simply the set Z2, the integers modulo 2. Suppose Aliceis given question x and returns answer a, and Bob is given question y and returns answer b. Therules of the game simply state that the players win precisely when xy = a + b. That is, the playerswin if the sum of their answers is congruent modulo 2 to the product of their questions.

If Alice and Bob are using a deterministic strategy for the CHSH game then they can’t hope towin with a probability greater then 0.75. If instead the players are allowed to form a strategy basedon the formalism of quantum mechanics then one can ask if the players can develop a strategythat wins the CHSH game with a probability greater then 0.75. This is exactly what is shown in[CHSH69] where the authors describe a quantum strategy for the CHSH game that can win with

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probability of approximately 0.85.This deep insight was able to propel the debate first started by Einstein, Podolsky and Rosen

from the setting of thought experiment to the setting of experimentally verifiable science. Theidea is to run a real life version of this game and try to experimentally demonstrate a strategythat wins with a higher probability than 0.75. Indeed, such experiments have been conducted andconsequently demonstrated a refutation of a classical description of the quantum world, see forexample [CS78].

Since the innovation of the CHSH game, many more non-local games have been studied in thecontext of quantum information theory. Examples include XOR games [CSUU07], linear constraintsystem games [CM12], graph parameter [PHMS19] and graph isomorphism games [AMR+19].The study of these games has wide ranging applications across many different fields of studythat intersect with QIT. As we have already remarked, non-local games play a central role intheoretical and experimental physics as they can be used to refute the possibility of a classicaldescription of quantum mechanics. Cryptographers can use non-local games to exploit the intrinsicrandomness of quantum mechanics to certify randomness for cryptographic protocols [VV12].Further applications are found in theoretical computer science, where non-local games can beused to define interesting complexity classes [CHTW04]. Surprisingly, non-local games have alsobeen shown to have powerful implications for deep problems within pure mathematics. This isprobably best illustrated by the recent resolution to the long standing Connes’ embedding problemfrom the theory of von Neumann algebras [JNV+20]. In this work the authors introduce a gamethat can distinguish between two different mathematical formalisms for quantum strategies ofnon-local games and hence resolve Connes’ embedding problem through a sequence of equivalentconjectures. For a good discussion on the connections between these equivalent conjectures see[Vid19].

1.1.2 Self-Testing

The fact that a simple game can be used to refute the possibility for a classical description ofquantum mechanics is indeed one of the crowning achievements of scientific discovery during the20-th century. This shows that if Alice and Bob use the building blocks of quantum mechanics,i.e., states and measurement operators, then they can outperform any deterministic strategy inthe CHSH game. A more subtle question to ask is what are the possible states and measurementoperators that Alice and Bob can use to win with the highest possible probability.

The mathematical formalism used to model quantum strategies for non-local games is describedin Chapter 2.6. In the context of the CHSH game, we know that each quantum strategy employedby Alice and Bob corresponds to a quantum state |ψ〉 in some Hilbert spaceH, and order 2 unitariesA0, A1 for Alice and B0, B1 for Bob acting onH. The condition of locality also requires that Alice’soperators commute with Bob’s. We consider the bias for this strategy, β, which is given by,

β = 〈ψ|A0B0 + A1B0 + A0B1 − A1B1|ψ〉. (1.1.1)

The probability of winning, ω, can be determined from the bias since β = 8ω− 4.The optimal winning value for the CHSH game can be shown to be 1

4 (√

2 + 2). One way thiscan be seen is through the use of sum of squares proofs. The theory of sums of squares techniques,and how they relate to non-local games more generally, is discussed in detail in Chapter 3.4. Using

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these techniques we can determine an upper bound on ω of 14 (√

2 + 2) since,

2√

2I − (A0B0 + A0B1 + A1B0 − A1B1) =

√2

4

(A0 + A1 −

√2B0

)2

+

√2

4

(A0 − A1 −

√2B1

)2≥ 0.

Thus we see that 2√

2 is an upper bound on the bias. Interestingly, in the case of the CHSH gamethere is a unique state and set of measurement operators that must be used in order to win with thehighest possible probability. The above sum of squares decomposition can also be used to showthis fact. Indeed, if this upper-bound is saturated with the state |ψ〉 then we have,

B0|ψ〉 =1√2(A0 + A1) |ψ〉, and B1|ψ〉 =

1√2(A0 − A1) |ψ〉. (1.1.2)

Using the above identities we can show that B0 and B1 must anti-commute with respect to |ψ〉:√

2 (B0B1 + B1B0) |ψ〉 = (B0(A0 − A1) + B1(A0 + A1)) |ψ〉= (A0 − A1)B0 + (A0 + A1)B1) |ψ〉= ((A0 − A1)(A0 + A1) + (A0 + A1)(A0 − A1)) |ψ〉= (A0A1 − A1A0 + A1A0 − A0A1) |ψ〉= 0.

Thus B0 and B1 anti-commute, that is −(B0B1)2 = I, with respect to a semi-norm induced by |ψ〉.

We can similarly show A0 and A1 anti-commute. These anti-commuting relations, along with thefact that B0 and B1 are order two unitaries, allow us to relate optimal quantum strategies to therepresentation theory for the group G, given by the following presentation.

G =⟨

X, Y, J |X2, Y2, J2, J(YZ)2, JXJ−1X−1, JYJ−1Y−1⟩

.

In particular if |ψ〉, A0, A1, B0, B1 is any quantum strategy that wins with a probability of 14 (√

2 + 2)then the map (X, Y, J) 7→ (A0, A1,−I) determines what we refer to as a |ψ〉-representation for G.For detail on |ψ〉-representations, please see Definition 5.2.2. Alternatively, a |ψ〉-representationcan be viewed as a representation of G, in the usual sense. If we let K denote the subspace of Hthat is generated by words in A0, A1 applied to |ψ〉, then the map (X, Y, J) 7→ (A0|K, A1|K,−I)determines a representation of G on K. A similar reasoning can be applied to the operators of Bob.Furthermore, there is a unique irreducible representation of G satisfying J 7→ −I given by

X 7→ σX =

[0 11 0

], Y 7→ σY =

[0 −ii 0

].

An application of Theorem 5.2.3 and Lemma 5.2.4 tells us that the operators of Alice and Bob, upto an application of an isometry, act as σX and σY on the state |ψ〉. We can also determine that thestate |ψ〉, up to an application of local isometry, is of the form |ψ〉′ ⊗ | junk〉. Where |ψ〉′ is given by,

|ψ〉′ = 1√4 + 2

√2

((1 +

1− i√2

)|00〉 −

(1 +

1 + i√2

)|11〉

).

This strategy can be seen to be unitarily equivalent to the more standard one presented withA0 = σX ⊗ I, A1 = σZ ⊗ I and |ψ〉 = |00〉+|11〉√

2. This shows that the CHSH game can be used to

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develop a protocol that can certify the maximally entangled state on two qubits and Pauli operatorsσX and σZ. Such certification protocols are often referred to as self-testing protocols and non-localgames that can be used for such protocols are called rigid.

The above discussion on the self-testing properties of CHSH is fairly technical and it is importantto not miss the forest for the trees here. This result tells us that if Alice and Bob want to employa strategy that wins the CHSH game with highest possible winning probability, then they aregreatly constrained in their choice of strategy. Firstly, we know that they must employ a quantumstrategy and take advantage of an entangled state. Furthermore, in a strong sense, there is a uniquequantum state and set of measurements that they must use. Astonishingly, all of this informationcan be determined by the exchange of classical information in form of the questions and answers.

More realistically, such a set up is unlikely to be realised with actual players named Alice andBob. Instead one can use these results to develop a protocol where by classical interaction with twoseparated black box quantum systems can certify the presence of a particular entangled state ormeasurement operators.

It is not hard to imagine that engineers and people in the applied sciences may one day make useof these protocols as important building blocks of technologies that exploit the nature of quantummechanics. Indeed, these protocols can allow scientists to certify what state they have their handson and what measurements were actually performed when working in lab.

Aside from the potential technological implications of such results, there is also an appealingaesthetic motivation here. Self-testing results, such as the one discussed above for CHSH, relyon the natural relations and symmetries of groups. In the case of CHSH, the proof requires theestablishment of the anti-commuting relations. These types of results allow for protocols whereone can certify the presence of group relations and symmetries in nature.

The main goal of the work contained in Chapter 5 was to explore several natural questionswithin the topic of self-testing. Precise formulations of these questions are detailed in Section5.1. We also refer the reader to this chapter for a discussion of what was previously known in theliterature regrading these questions.

Question 1.1.1. What states can we self-test for using non-local games?

Question 1.1.2. What operators can we self-test for using non-local games?

Question 1.1.3. Can we use non-local games to certify the presences of certain algebraic relations amongstoperators?

Question 1.1.4. In what ways is it possible for a non-local game to fail to provide a self-test protocol for anyset of operators or state?

Several of the above questions were suggested to the authors of [CMMN20] by Professor HenryYuen at the outset of a summer research project in 2019. This research project would eventuallybecome the work presented in Chapter 4. Although the work does not provide a complete answerto any of the above questions, it does provide a substantial addition to what is known to thescientific community regarding these questions. In order to obtain these results, we introduce anew family of non-local games that can be realised as generalizations of the CHSH game to Zn, thering of integers modulo n. Different generalizations of the CHSH game have been studied beforebut proved to be less amenable to analysis [BS15]. We show that this family of games all havequantum strategies that outperform the best classical strategy. We obtain our results on self-testingby employing a collection of techniques sometimes referred to as sum of square proofs. Combiningthese techniques with representation theory of finite groups and a powerful theorem due to Gowersand Hatami [GH17] we are able to obtain our results on the topic of self-testing for our game in the

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case n = 3. Since Chapter 5 already contains an extensive introduction we direct readers to 5.1 formore detailed discussion of this work.

1.2 Quantum Graphs

Chapter 4 focuses on a collection of results on a topic sometimes referred to as quantum graphtheory. Quantum graph theory is an operator space generalization of graph theory that was initiallymotivated by work both in complexity theory [SB07] and physics [DSW13]. In order to understandthese motives, we first discuss the well known connections between noisy channels, graph theory,and complexity theory. We also note that quantum graph theory has appeared in the literature asnon-commutative graph theory, which is the name that is used in Chapter 4.

1.2.1 Noisy channels and graphs

Suppose Alice uses a finite alphabet A = a1, a2 . . . an to send messages through a noisy channel toBob, who receives them as letters from his finite alphabet B = b1, b2, . . . , bm. We can representthe noisy channel by a matrix N = (P(bi|aj)), where P(bi|aj) determines the probability that Bobreceives letter bi given that Alice sent aj. The study of the zero-error capacity of such noisy channelswas first due to Shannon [Sha56]. Here we define the zero-error capacity of channel N as themaximum number of ai ∈ A that Alice can send through the channel such that Bob knows exactlywhat was sent. In other words, this is the size of the largest collection ai1 , . . . aim ∈ A such that forany bk ∈ B we have p(bk|aij)p(bk|aik) = 0 for all choices of j 6= k.

The study of zero-error capacity and similar notions of channel capacity are important forinformation theory and have led to foundational results such as Shannon’s noisy channel codingtheorem [Sha48]. Graph theory is an important tool in this analysis. For our purposes we will onlyconsider loop-free, undirected graphs. A graph G will be a an ordered pair G = (V, E) consistingof a finite set of vertices V and E ⊂ V×V such that if (ei, ej) ∈ E then i 6= j and (ej, ei) ∈ E. Given anoisy channel N = (P(bi|aj)) we can construct an associated graph, GN , known as the confusabilitygraph of the channel. This graph will have vertices consisting of Alice’s alphabet, V = A, and fori 6= j we have (ai, aj) ∈ E precisely when P(bk|ai)P(bk|aj) 6= 0 for some bk ∈ B. It is not hard tomake the observation that the independence number, α(GN), is equal to the zero error capacity of thenoise channel N.

Another important parameter of a noisy channel N is the so called least packing number. Supposein addition to her noisy channel N, Alice also has a separate, expensive, yet noise free channel Mthat she can use to send information to Bob. The least packing number of N tells us how much extrainformation Alice needs to send through M such that she can use her entire alphabet A without thepotential for any errors. More formally, suppose Alice has two finite alphabets A and C that shecan send through noisy channel N = (P(bi|aj)) and noise free channel M, respectively. Alice sendsinputs (ai, cj) through N ×M to Bob, who receives (N(ai), cj). The least packing number is thesmallest size of alphabet C such that Bob can always identify ai from (N(ai), cj). One can determinethis value by taking the chromatic number, χ(GN), of the associated confusability graph G.

Aside from the connections to information theory, the parameters α(G) and χ(G) are alsointeresting from the point of view of complexity theory. The complexity class NP consists of theclass of problems whose solutions can be verified by a polynomial-time deterministic verifier. Aproblem is called NP-hard if it is as difficult to solve as any NP problem. The study of NP-hardproblems was initiated by Cook [Coo71], Levin [Lev73] and Karp [Kar72]. Calculating the graphparameters α(G), and χ(G) are both known to be NP-hard. Consequently, the same can be saidabout calculating the zero-error capacity or least packing number of a noisy channel. There is no

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known method to efficiently calculate either of these parameters. In [Lov79] Lovász introduces asemi-definite programming relaxation, which we call θ(G), which can be used to provide boundson these parameters. The relationship is summarized below in what is sometimes referred to asthe Lovász sandwich theorem, α(G) ≤ θ(G) ≤ χ(G). Here G denotes the graph complement of G,which is obtained from G by having an edge between disconnected points in G and removing anyedge between connected points in G.

1.2.2 Quantum channels and quantum graphs

Informally, a quantum channel is a linear map between two matrix algebras that both preserves thetrace of a matrix and preserves positivity of a matrix, in a strong sense. More precisely, in Chapter 4we view quantum channels as completely positive, trace preserving maps. Early work on quantumgraph theory has had several motives. In [SB07] Shor and Beigi introduce a notion of zero errorcapacity for a quantum channel and show that computing this capacity is QMA-hard. QMA is theclass of problems for which there exists a short “quantum proof” or a quantum state, that can beverified with high probability by a “quantum verifier". It is considered to be the analogue of thecomplexity class NP for quantum computers. This result can be viewed as a quantum version ofthe fact that it is NP−hard to compute the zero error capacity of a noisy channel.

In [DSW13] authors Duan, Severini and Winter further expand on the topic of error capacity for aquantum channel. Instead of having motives focused on complexity theory they generalize the notionof a confusability graph to the setting of quantum channels. In particular they show that to eachquantum channel N : Mn → Md one can associate to it a special type of linear subspace or operatorsystem, SN ⊆ Mn. The relationship between a quantum channel, N , and the operator system , SN ,is analogous to the relationship between a noisy channel and the associated confusability graph. Itis for this reason that operator systems are sometimes referred to as quantum graphs. Furthermore,Duan, Severini, and Winter define the notion of an independence number, α(S), and parameter,θ(S), for an operator system. They are able show that the zero error capacity of a quantum channel,N , is equal to the independence number, α(SN ), of it’s associated quantum graph. They also showthat these parameters generalize the first part of the Lovász sandwich theorem as they satisfy theinequality α(S) ≤ θ(S).

In addition to the two motivations above quantum graphs also prove to be of great mathematicalinterest. It possible to view any noisy channel N as a special case of a quantum channel and it is alsopossible to view any graph G as a special case of a quantum graph SG. For the latter constructionrefer to Definition 4.2.1 in Chapter 4. The work of Paulsen and Ortiz tell us that the inclusion ofgraphs as quantum graphs via G 7→ SG preserves all the graph theoretic information [OP15]. It isdue to these observations that the study of quantum graphs should properly be thought of as amathematical generalization of graph theory instead of simply an analog for it. Graph theory is anenormous topic and is central to the study of combinatorics. In light of the work of Paulsen andOrtiz one can generate questions of mathematical inquiry by exploring which graph theory resultsextend to the more general setting of quantum graphs.

There are several different motivations for the study of quantum graphs, including mathematicalmotivations as well as motivations from complexity theory and physics. As the different motives forstudying quantum channels are pursued it would be naive to assume that results which successfullygeneralize ideas from one area will always be of significance for all. Indeed, it is surprising thatin the case of the independence number of a quantum graph, all three of these goals are aligned.From a mathematical stand point we have α(G) = α(SG) for all graphs G. Additionally α(S) bothextends the notion of NP-hard to QMA-hard and extends the notion of zero error capacity of anoisy channel to the zero error capacity of a quantum channel. It is not inconceivable that one

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could define a parameter β such that α(G) = β(SG) for all graphs but it is neither QMA-hard tocompute or able determine the zero error capacity of a quantum channel.

1.2.3 A Quantum Lovász inequality

One of the primary goals of the work in Chapter 4 was to investigate in what sense one can properlygeneralize the inequality, θ(G) ≤ χ(G), which is the second half of the Lovász sandwich theorem.

In order to investigate this we were motivated to generalize the notion of graph complementG 7→ G in the setting of quantum graphs. The natural map to try is to send a quantum graphS ⊆ Mn to it’s orthogonal complement S⊥ under the Hilbert-Schmidt inner product on Mn. Sinceevery operator system will always contain the identity matrix the map S 7→ S⊥ seemingly failsto send a quantum graph to a quantum graph. Our approach was to also consider a notion ofquantum graph first introduced by Stahlke [Sta16] that is similar but distinct to quantum graphs asconceived in [DSW13]. We refer to the two notions as submatricial traceless self-adjoint operator spaceand submatricial operator system respectively. In Theorem 4.2.7 we establish that the relationshipbetween the two notions is precisely that of orthogonal complementation under the Hilbert-Schmidtinner product and we use this fact to extend the notion of graph complements to quantum graphs.The second ingredient needed to generalize the inequality θ(G) ≤ χ(G) is a notion of chromaticnumber of a quantum graphs.

In [HPP16] the notion of the least packing number of a quantum channel is described. Fur-thermore, they define a quantum graph parameter χ(S) that both satisfies χ(G) = χ(SG) and cancalculate the least packing number of a quantum channel. Due to these desirable properties thismay be a suitable choice for the chromatic number of a quantum graph. Unfortunately, when itcomes to the goal of generalizing the Lovász sandwich theorem this parameter falls short. Indeedin Example 3 we provide an operator system S such that θ(S) = n > χ(S⊥) = 1. This tells usthat, in this case, one can not hope to satisfy all of the competing motives for quantum graphs. Inparticular, if one wants to keep a connection to information theory via the least packing number,then one can not hope to also generalize the Lovász inequality.

In Definition 4.3.1 we introduce a new parameter which we call the the strong chromatic number.In Theorem 4.3.7 we show that this parameter can be used to successfully generalize the secondhalf of the Lovász sandwich theorem. Of-course, as discussed in the previous paragraph we cannothave our cake and eat it too. The strong chromatic number fails to have any straight forwardconnection to computing the least packing number of a quantum channel or any other propertyrelevant to information theory. It is for this reason that the work in Chapter 4 focuses entirely onquantum graphs as mathematical generalizations of graphs and does not mention noisy channelsor quantum channels.

An alternative approach to generalizing Lovász sandwich theorem was first established byStahlke in [Sta16]. Stahlke makes connections to quantum channels much more the focus of hiswork and defines a chromatic number for quantum graphs which we denote χSt. We provide anupper and lower bound for χSt.

1.2.4 Sabidussi Theorem and Hedetniemi’s Conjecture

Graph theory is an area of mathematics where simple to state problems can often prove to bequite intractable. This is perhaps best illustrated by the famous Four Color Theorem, whichtook over 100 years from its conjecture to its eventual resolution by Appel and Haken [AH89].Many other outstanding questions involving the colourings of graphs remain open, includingthe Erdos–Faber–Lovász conjecture and Hadwiger’s conjecture. If a full resolution to a problem

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remains intractable incremental progress can be made by either attempting to prove the result in amore narrow setting or refuting a version of the conjecture in a more general setting. Quantumgraphs can provide a more general setting for which one can test conjectures against. We wereinitially motivated to provide a more general setting for the refutation of the longstanding Hedet-niemi’s conjecture. At the time of our earliest pre-print this conjecture remained unsolved [KM17].Surprisingly, this question has since been resolved in the negative by the remarkable work of Shitov[Shi19].

In order to help introduce Hedetniemi’s conjecture we consider the following outlandish sci-fiscenario:

A deadly disease is spreading world-wide and all physical access to the local math department hasbeen halted. Virtual classrooms, where graduate students can congregate, are to be created. Tofoster harmonious social interaction graduate students will be grouped together provided that

either they have complementary math specialties, or they pursue complementary hobbies.

In such a far-fetched scenario what is the least number of virtual classrooms that would be needed?In order to answer this question one could generate two graphs. The first graph would consist ofthe math specialties of all students, and would have an edge between two specialties if they arenot complimentary, such as axiomatic set theory and applied fluid dynamics. The second graphwould consist of graduate student hobbies and would link two hobbies provided they are notcomplimentary, such as crocheting and gunsmithing. If we let k be the lesser of the chromaticnumbers of the two graphs then k virtual rooms will be sufficient. Hedetniemi’s conjecture purportsthat k will always be the least number of rooms needed. More formally, if we let G× H denote thecategorical product of two graphs, then the following equality always holds,

χ(G× H) = minχ(G), χ(H).

One direction of this inequality is fairly trivial and the conjecture speculated as to the other. Thisseemingly innocuous statement remained unsolved for half a century. In order to formulate aversion of Hedetniemi’s conjecture for quantum graphs we generalize the categorical product oftwo graphs to the setting of quantum graphs. Given two graphs G and H one can construct a thirdgraph known as the categorical product, G× H, of G and H as follows. The nodes of G× H willconsist of all order pairs (v, a) where v is a node from G and a is a node from H. We say thereis an edge between nodes (v, a) and (w, b) whenever there is an edge between v and w in G andan edge between a and b in H. For any graphs G and H a coloring of either graph can be usedto construct a coloring for the categorical product G × H. Consequently one has the followinginequality χ(G× H) ≤ minχ(G), χ(H). The question as to whether or not the above is actuallyan equality for all graphs G and H was the long standing conjecture due to Hedetniemi [Hed66].We call the above inequality, Hedetniemi’s inequality, and in Section 4.4 we obtain a generalizationfor this in the context of quantum graphs. We note that the reverse inequality has been shown tonot hold for all graphs [Shi19].

A similar identity exists for the Cartesian product and is due to Sabidussi [Sab57]. The Cartesianproduct of two graphs G and H, written GH, is given as follows. The nodes of GH will consistof all order pairs (v, a) and there will be an edge between (v, a) and (w, b) if either there is an edgebetween v and w and a = b or v = w and there is an edge between a and b. Sabidussi’s theoremstates that for any two graphs G and H we have χ(GH) = maxχ(G), χ(H). In Section 4.4 wegeneralize this identity to the setting of quantum graphs. We achieve this generalization for boththe strong chromatic number and an additional parameter we introduce referred to as the minimalchromatic number.

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1.2.5 Future directions

Exhibiting an example of a QMA-hard problem was one of the earliest motivations for thinkingof quantum graphs. One opportunity for further investigation is to determine if any of theabove mentioned chromatic number graph parameters for quantum graphs are also QMA-hard tocompute. This would be analogous to the fact that both the independence number and chromaticnumber of a graph are NP-hard to compute. Although we do not investigate this, in Theorem 4.2.15and Theorem 4.3.5 we show how to compute various parameters for quantum graphs in terms ofcomputing their classical analog across families of graphs.

Several other works have been completed that focus on extending notions from graph theory tothe setting of quantum graphs. In [Wea17] Weaver examines extremal graph theory in the setting ofquantum graphs and successfully obtains a generalization of a well known theorem due to Ramsey.A version of Ramsey’s theorem is obtained for quantum graphs as infinite dimensional operatorsystems in the work of Kennedy et al [KKS17]. The study of graphs that have infinite nodes hasbeen a part of graph theory since it’s inception featured by König’s in his 1936 book Theorie derendlichen und unendlichen Graphen (The Theory of finite and infinite graphs). For an updatedversion featuring commentary by Tutte please see [Kö12]. The study of operator systems in infinitedimensions have provided a rich landscape of theory and structure. The work of Kennedy etal. shows that there is much potential in connecting ideas from infinite graph theory to operatorsystems on infinite dimensional spaces.

Another setting where ideas from graph theory heavily intersect with work in quantum infor-mation theory is in the study of non-local games. In [CNM+07] the notion of a quantum chromaticnumber is introduced. Here two isolated players try to convince a verifier that a given graph isk−colourable. If the players have access to entanglement then it is possible to win this game evenif the given graph is not k−colourable. The study of non-local games based on graphs has grownconsiderably and the literature also features games based on the independence number, graphhomomorphism and graph isomorphism. For some select highlighted works see [PT15], [RM16],[SSTW16]. In [SBG20] a non-local game is introduced that is based in part on homomorphismsbetween quantum graphs. This work provides motive for further exploration of non-local games.

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Contents

1 Introduction 11.1 Non-locality and Self-testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Non-Local Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Self-Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Quantum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Noisy channels and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Quantum channels and quantum graphs . . . . . . . . . . . . . . . . . . . . . 71.2.3 A Quantum Lovász inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Sabidussi Theorem and Hedetniemi’s Conjecture . . . . . . . . . . . . . . . . 81.2.5 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Operator Theory Formalism of Quantum Mechanics 132.1 The Axioms of a Quantum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Classical vs. Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 States and Observables as Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Quantum Strategies for non-local games . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Sums of Squares 253.1 Non-commutative Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Sums of Squares for Group Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Generalized Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 Approximation Properties of the Ring of Integers . . . . . . . . . . . . . . . . 28

3.3 A Weak Sum Of Squares Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Non-local games and Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Groups and Commuting Operator Strategy Correspondence . . . . . . . . . . 323.4.2 Sum of Squares Certificates for Games . . . . . . . . . . . . . . . . . . . . . . . 33

4 Quantum Graphs 354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Non-commutative graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.1 Non-commutative graphs as operator systems . . . . . . . . . . . . . . . . . . 374.2.2 The complement of a non-commutative graph . . . . . . . . . . . . . . . . . . 38

4.3 Non-commutative Lovász inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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4.3.1 The Strong chromatic number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.2 The minimal chromatic number . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Sabidussi’s Theorem and Hedetniemi’s conjecture . . . . . . . . . . . . . . . . . . . . 454.4.1 Sabidussi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.2 Hedetniemi’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Non-local Games 495.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.3 Relation to prior generalizations of CHSH . . . . . . . . . . . . . . . . . . . . 545.1.4 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.5 Organization of paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2.2 Non-local games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.3 Linear constraint system games . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.4 Gowers-Hatami theorem and its application to self-testing . . . . . . . . . . . 59

5.3 A generalization of CHSH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Strategies for Gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.1 Definition of the strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4.2 Analysis of the strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.5 Group structure of Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 Group structure of Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.7 Optimality and rigidity for G3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.7.1 Optimality of S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.7.2 Algebraic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.7.3 Rigidity of G3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.8 SOS approach to solution group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.9 A non-rigid pseudo-telepathic LCS game . . . . . . . . . . . . . . . . . . . . . . . . . 85

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Chapter 2

Operator Theory Formalism of QuantumMechanics

The early development of quantum mechanics is not attributable to any single scientist as it is theculmination of many notable contributing physicists including Max Planck, Albert Einstein, NielsBohr, Erwin Schrödinger and Werner Heisenberg. Similarly, the mathematical formalism that waseventually able to provide a unifying framework for quantum mechanics is due to several notablecontributors including David Hilbert, John von Nuemann, Francis Joseph Murray and Paul Dirac.Most introductory courses in quantum information theory take for granted this operator theoryformalism without providing much insight on it’s origins.

In this section we develop this formalism starting from a collection of axioms detailing theprobabilistic nature of quantum states and their accompanying observables. Discussions of physicalintuition or experimental motivation is not a primary motivation here. Instead, we aim to presentthis content in a style that resembles a graduate course in abstract mathematics. That is, as oftenas possible, we explicitly state definitions and clearly state important conclusions as theoremstatements or exercises. The content presented here was initially completed during a graduatereading course on The Mathematical Foundations of Quantum Mechanics by George W. Mackey and islargely based on Chapter 2-2 [Mac04].

2.1 The Axioms of a Quantum System

We view a quantum system S = (O,S) as an ordered pair comprised of states S and the set ofobservables O. To each pair consisting of an observable A ∈ O and state α ∈ S , there exists aBorel probability measure, αA on the real line R. Informally, this means each observation we canmake about a state is fundamentally random. Furthermore, the associated probabilities of ourobservations are determined by both what we are trying to observe and what state we are observing.The description above is far too general as it says little more then observations of states are random.If we instead require this family of probability measures to satisfy a handful of somewhat naturalaxioms then a rich mathematical structure emerges. We present the defining axioms of a quantumsystem in Definition 2.1.1.

Before we proceed to the definition we need to recall some notation regarding probabilitymeasures. Given a probability measure p and Borel function f : R → R we can define thepush-forward measure, denoted f (p), by f (p)(E) = p( f−1(E)). It is clear that f (p) will also bea probability measure. Also if pi is a sequence of probability measures and ti are non-negativenumbers such that ∑i ti = 1 then we can define a new probability measure p = ∑i ti pi by p(E) =

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∑i ti pi(E) for all E ⊆ R. For any Borel set E we can define the indicator, or characteristic functionχE, which evaluates to 1 on the set E and 0 elsewhere.

Definition 2.1.1. A quantum system S is an ordered pair, S = (O,S), consisting of a set of observ-ables O, and states S , such that for each pair α ∈ S and A ∈ O, there exists a Borel probabilitymeasure αA on R, and this family of measures satisfies the following axioms:

1. If αA = αA′ for all α ∈ S then A = A′. If αA = α′A for all A ∈ O then α = α′.

2. For each A ∈ O and for each Borel function f , there exists a unique observable f (A) ∈ Osuch that α f (A) = f (αA) for all states α ∈ S .

3. If αii∈N is a sequence of states and tii∈N is a sequence of non-negative real numbers suchthat ∑i ti = 1, then there exists a unique state α ∈ S such that for all A ∈ O we have

αA = ∑i

ti(αi)A.

In order to state the last two axioms for Definition 2.1.1 we introduce some notation regarding aparticularly important set of observables called questions. An observable Q ∈ O is called a questionif αQ(0, 1) = 1. We let Q ⊆ O denote the set of all questions and write Q1 ≤ Q2 if αQ1(1) ≤αQ2(1) for all α ∈ S . We also say questions Q1 and Q2 are disjoint if αQ1(1) + αQ2(1) ≤ 1 forall α ∈ S . Additionally, a map q from Borel sets B to Q is called a question valued measure if:

a) E⋂

F = ∅ =⇒ q(E) and q(F) are disjoint questions.

b) Eii are pairwise disjoint then q(⋃

i Ei) = ∑i q(Ei)

c) αq(∅) = δ0 and αq(R) = δ1 for all α ∈ S .

Keeping the notation outlined above we now return to our axioms.

4. Let Qii∈N be a sequence of pairwise disjoint questions. Then there exists a questionQ = ∑i Qi such that for all α ∈ S

αQ(1) = ∑i

αQi(1).

5. If q is a question valued measure then there exists an observable A ∈ O such that q(E) =χE(A) for all E ∈ B.

The first axiom simply tells us that states and observables are exactly determined by the proba-bility measures they can generate. Informally, the second axiom says that the set of observationswe can make is closed under the application of any reasonable function and the third axiom tells usthat the set of states is closed under infinite convex combinations. Non-trivial convex combinationsof states are viewed as mixtures of states with state αi occurring with probability ti.

In the above definition of a quantum system we also smuggled in some important concepts.In particular we defined the set of observables Q ⊆ O which we call questions. Questions can bethought of as observables that receive a “yes” or “no” response when an observation is made ofany particular state. We equipped this set with a partial order where one question is greater thananother if the probability of a “yes” response is greater for every state.

We also can equip the set of questions with a orthocomplement by letting the complement of aquestion Q to be the question 1− Q. Where 1− Q is satisfies α1−Q(1) = 1− αQ(1 for all

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states α. By Axiom 1 this is enough to uniquely define the the question 1− Q. More simply, thecomplement of a question Q is a question 1− Q that receives a “yes” response with the sameprobability that question Q receives a “no” response. Partial motivation for the the notion ofdisjoint questions is given in Exercise 2.1.3 below. It should be noted that this definition of disjointquestions allows for the possibility of a question to be disjoint from itself. Indeed, as long asαQ(1 ≤ 1

2 this will be the case.The set of questions together with their partial order and orthocomplement will play a central

role in the rest of our discussion. In fact, we will eventually see that they are sufficient to recover theentire quantum system S = (O,S). Below are a collection of exercises that provide some insightinto the structure of questions and provide some mathematical motivation for the partial orderingand the orthocomplement.

Exercise 2.1.1. An observable A ∈ O is a question if and only if A = A2. (Here we use the notation A2 todenote the image of A under the Borel function x 7→ x2.)

Exercise 2.1.2. For each Borel set E ∈ B and observable A ∈ O there exists a unique question χE(A) ∈ Qthat satisfies

αχE(A)(1) = αA(E)

αχE(A)(0) = αA(R\E)

for all states α ∈ S .

The question χE(A) can be understood as the question that receives a “yes” response with thesame probability that A makes an observation in the set E.

Exercise 2.1.3. Let E, F ∈ B be Borel sets. Then for any observables A ∈ O the following are true:

• If E⋂

F = ∅ then χE(A) and χF(A) are disjoint questions.

• If E ⊆ F then χE(A) ≤ χF(A).

• χE(A) is the orthocomplement of χE(A).

Mackey also provides a way to view the question Q in Axiom 4 as the least upper-bound ofQi, which the following theorem outlines.

Theorem 2.1.2. If Qii is a sequence of pairwise disjoint questions and R ≥ Qi for all i ∈ N, thenR ≥ ∑i Qi.

Proof. The proof for this result is given in [Mac04]. Alternatively, it may be considered as a exercise,although it is quite challenging.

Thus far it may seem that in a quantum system S = (O,S), the sets of observables O and statesS are not very intuitive. We conclude this section by discussing a correspondence to these sets thatmakes working with them more natural. First, we note that there is a one-to-one correspondencebetween the set of observables O and the set of question valued measures q : B→ Q.

Theorem 2.1.3 (Observables correspond to question valued measures). For each question valuedmeasure q the observable A defined from Axiom 5 is unique. Moreover, without loss of generality one canassume there is a one-to-one correspondence between the set of observables O and the set of question valuedmeasures.

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Proof. the first part of this proof is suitable for an exercise. For the second please consult Makey’sdiscussion.

In order for us to determine a useful correspondence for the set of states S , we need to definean abstract state on the set of questions.

Definition 2.1.4. We say a map m : Q → [0, 1] is called an abstract state on the set of questions if itsatisfies the following criteria:

a) If Qi are pairwise disjoint then m(∑i Qi) = ∑i m(Qi).

b) We will simply use 0 to denote the question satisfying α0 = δ0 for all α ∈ S . Similarly, we let1 denote the question satisfying α1 = δ1 for all α ∈ S . Then we require the map m to satisfym(0) = 0 and m(1) = 1.

We note that Mackey refers to these maps as "probability measures on the set of questions".Next are a series of exercises that show given any state α ∈ S , we can obtain an abstract state on

the set of questions. We also see that this set is strongly convex in the sense that is is closed undercountable convex combinations.

Exercise 2.1.4 (States give rise to abstract states). If α is a state, show that the map mα : Q → [0, 1]defined by mα(Q) = αQ(1) is an abstract state on the set of questions. Moreover, show that if mα = m′αthen α = α′.

Exercise 2.1.5. Let αii∈N be a sequence of states and tii∈N a sequence of non-negative real numberssuch that ∑i ti = 1. Show that m = ∑i timαi defines an abstract state and that m∑i αi = m.

We will eventually see that the set of abstract states and the set of states are in one-to-onecorrespondence. One of the directions of this correspondence was outlined in Exercise 2.1.4. We donot make the other direction automatic by the addition of an axiom as was the case for obervablesvia axiom 5. For the moment, we remark that if there was a one-to-one correspondence, then theset of abstract states on the set of questions would have to satisfy the two following properties:

1. If mii is a sequence of abstract states and tii are non-negative numbers that sum to 1,then there exists an abstract state m such that m = ∑i timi.

2. If m(Q1) ≤ m(Q2) for all abstract states m then Q1 ≤ Q2.

A collection of abstract states on the set of questions that satisfy Property 1 is called strongly convexand we say that the set is full if it satisfies Property 2.

We will return to the discussion of abstract states on the set of questions later. Our nextimmediate goal is to show that a quantum system can be completely determined by the partiallyordered set of questions Q. To do this, we need to introduce the notion of a quantum logic.

2.2 Quantum Logic

We will eventually show that one can determine the entire quantum system S = (O,S) from the setof questions Q. In order to do this, we introduce the idea of a quantum logic, which is a partiallyordered set together with special involution called an orthocomplement. We show that given anyquantum logic, one can determine a set of observables O and states S that satisfy the axioms fromDefinition 2.1.1. Conversely, given any quantum system S = (O,S) one can determine a quantumlogic by looking at the set of questions Q ⊆ O.

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Definition 2.2.1. A quantum logic is a partially ordered set L equipped with an an involution a 7→ a′

such that:

1. If a1 ≤ a2 then a′2 ≤ a′1

2. If aii is a sequence of elements of L such that ai ≤ a′j for i 6= j, then there exists a uniqueleast upper bound for this sequence, denoted

⋃i ai.

3. a ∪ a′ = b ∪ b′ for all a, b ∈ L. We call this element 1.

4. If a ≤ b then b = a ∪ (b′ ∪ a)′ for all a, b ∈ L.

Different definitions for a quantum logic than the one above appear in mathematical literature.The partial order and involution of a quantum logic should remind us of the set of questions

in a quantum system. We say that elements a1 and a2 are disjoint if a1 ≤ a′2 and we refer tothe involution with the above properties as an orthocomplement on L. As we did for the set ofquestions in a quantum system, we can define the notions of an abstract state and aa L−valuedmeasure for a quantum logic.

Definition 2.2.2. A function m : L → [0, 1] is an abstract state on L if

a) m(1) = 1 and m(1′) = 0.

b) m(⋃

i ai) = ∑i m(ai) for any sequence of disjoint elements ai.

A family F of abstract states on L is called full if m(a1) ≤ m(a2) for all m ∈ F implies thata1 ≤ a2. A family F is called strongly convex if it is closed under infinite convex combinations.

Definition 2.2.3. An L-valued measure is a function q : B→ L that satisfies

a) E ∩ F = ∅ =⇒ q(E) and q(F) are disjoint elements of L.

b) If Eii are pairwise disjoint then q(⋃

i Ei) =⋃

i q(Ei).

c) q(∅) = 1′ and q(R) = 1.

In order to see how a quantum logic can give rise to a quantum system, we need a naturalway of constructing probability measures. Given an L-valued measure q and an abstract statem : L → [0, 1], we define a Borel probability measure p by p(E) = m(q(E)). We are now ready tostate the main theorem of this section.

Theorem 2.2.4. If S = (O,S) is a quantum system then the partially ordered set of questions equippedwith the involution Q 7→ 1−Q is a quantum logic. Conversely, if L is a quantum logic, we can obtain aquantum system by taking the observables O to be the set of L-valued measures and take the set of states Sto be any full and strongly convex set of abstract states.

Proof. Suppose S = (O,S) is a quantum system. We show the involution Q 7→ 1− Q satisfiesthe four properties in Definition 2.2.1. If Q1 ≤ Q2 then, by definition, for all states α we haveαQ1(1) ≤ αQ2(1). So for any state α we have α1−Q1(1) ≥ α1−Q2(1), thus 1−Q1 ≥ 1−Q2.The second property follows from the Axiom 4 together with Theorem 2.1.2. Here, the sum ofdisjoint questions takes the role of the least upper bound. Properties 3 and 4 can be shown to holdas well.

The converse is left as an exercise to the reader.

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When viewing a quantum system as a quantum logic, we see that observables are determinedby the set of L-valued measures and states are determined by a full and strongly convex family ofabstract states. We eventually want to see how we can view the set of observables as self-adjointoperators on a separable Hilbert space and states as density operators. Before moving to thisdiscussion, we take a small detour to talk about the differences between classical and quantumsystems.

2.3 Classical vs. Quantum Systems

One crucial part of obtaining a good mathematical foundation for quantum systems is being able todifferentiate them from classical systems. In classical systems, there is a one-to-one correspondencebetween questions and Borel subsets of a symplectic manifold. In order to understand how thisview fails in the generality of quantum systems, we introduce the idea of simultaneously answerablequestions.

Definition 2.3.1. Let Q1 and Q2 be questions. We say Q1 and Q2 are simultaneously answerable(or compatible) if there exists mutually disjoint questions Q3, R1, R2 such that Q1 = R1 + Q3 andQ2 = R1 + Q3. We also say two observables A and B are compatible if for all Borel sets E, F thequestions χE(A) and χF(B) are compatible.

The idea of compatible questions is both intuitive and natural. Informally, when two questionsare compatible then it makes sense to ask about the probabilities that both events occur and theprobability that one but not the other event occurs. See Figure 2.1 to help motivate the definitionof compatible questions. It is the departure from this intuitive picture that is the main divergencebetween classical and quantum systems.

R1 R2Q3

Figure 2.1: Two compatible questions Q1 and Q2.

Exercise 2.3.1 (Classical Comparison). If Q1 and Q2 are any classical questions (i.e Borel subsets of asymplectic manifold) then they are simultaneously answerable.

Exercise 2.3.2. Questions Q1 and Q2 are simultaneously answerable if and only if there is an observable Aand Borel sets E1 and E2 such that Q1 = χE1(A) and Q2 = χE2(A).

The axioms we have presented thus far do not imply that that every two questions are simulta-neously answerable. In the situation that any two questions are compatible then the set of questionscan be shown to form a Boolean algebra.

Definition 2.3.2. A partially ordered set, equipped with the usual union and intersection operationsaswell as with a orthocomplement is called a Boolean algebra if it is a lattice (i.e any two elementshave a least upper-bound and greatest lower-bound) and satisfies the distributive law (i.e Q1 ∩(Q2 ∪Q3) = Q1 ∩Q2 ∪Q1 ∩Q3).

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Exercise 2.3.3. If any two questions are simultaneously answerable then the set of questions Q is aBoolean algebra. Conversely, if the set of questions is a Boolean algebra with the canonical ordering andorthocomplement then any two questions are compatible.

The conclusion allows us to highlight the difference between classical systems, which corre-spond to Boolean algebras, and more general quantum systems, which correspond to what wedefined as a quantum logic in subsection 2.2. We next will see how quantum systems can be realisedby looking at Hilbert spaces.

2.4 Hilbert Spaces

In this subsection, we outline the connection between the definition of a quantum system as givenin Definition 2.1.1 and the more common setting of a quantum system on a separable Hilbert space.The way this connection is made is to recognize that one can obtain a quantum logic from lookingat the collection of closed subspaces of a Hilbert spaceH. This collection is equipped with a partialorder given by containment. We can also use the orthogonal complement given by the Hilbertspace inner product to define an orthocomplement in the sense of Definition 2.2.1. We leave thesimple details to the reader.

Theorem 2.4.1. The partially ordered set of all closed subspaces of a separable Hilbert spaceH is a quantumlogic and thus a realisation of a quantum system S = (S ,O).

Proof. Exercise.

Mackey treats this discussion differently. First, he includes a lengthy discussion about possiblyweakening the definition of Boolean algebras which we completely omit. Second, he notes thatthe realisation of a quantum system in terms of the partially ordered closed subspaces of a Hilbertspaces is one of many possible choices and includes this particular choice as an additional axiom.Please refer back to Mackey to see his comments on this choice. Due to the following theorem byKakutani, we do not need to consider the more general setting of Banach spaces.

Theorem 2.4.2 (Kakutani). If the lattice of closed subspaces of a Banach space X is equipped with anorthocomplement then the norm is equivalent to a Hilbert space norm and the orthocomplement is thestandard one on a Hilbert space.

A valid alternative to the Hilbert space approach is to consider the lattice of projections in afactor von-Neumann algebra; we do not consider this possibility here. Instead, we restrict ourselvesto those quantum systems S = (S ,O) that can be determined by the set of closed subspaces ofsome separable Hilbert spaceH. Using the one-to-one correspondence between closed subspacesand orthogonal projections P ∈ B(H), we can view the set of questions Q ⊆ O as being the setof projection operators on the Hilbert space H. Of course, since B(H) is a factor von-Neumannalgebra, this can be viewed as a special case of the von-Neumann algebra approach.

Many familiar properties and definitions pertaining to operators on a Hilbert space can beextended to a quantum system in a meaningful way. Before we move towards our final goal ofcharacterizing observables and states as particular operators, we look at some of these definitions.For the next few definitions, we fix a quantum system S = (S ,O) in the sense of Definition 2.1.1.

Definition 2.4.3. Let A ∈ O be an observable and E ⊆ B be a Borel set. We say E is of measure zerowith respect to A if for all states α ∈ S we have αA(E) = 0. We also let ρ(A), called the resolvent ofA, denote the union of all open intervals I that are of measure zero with respect to A and let σ(A),called the spectrum of A, denote the complement of ρ(A).

19

Exercise 2.4.1. E is of measure zero with respect to A if and only if χE(A) = 0.

Definition 2.4.4. The set of points x ∈ H such that χx(A) 6= 0 is called the point spectrum of A.If the spectrum of A is a bounded set then we say A is bounded. The norm of A is defined as‖a‖ = supλ∈σ(A) |λ|.

Definition 2.4.5. If A is bounded then for each α ∈ S , the measure αA is concentrated in a finiteinterval and hence we can evaluate the integral∫ ∞

−∞xαA.

We denote this value mα(A) and call this the expected value.

Exercise 2.4.2. If the observable A is a question then A is bounded and mα in the definition above agreeswith definition from Exercise 2.1.4.

Exercise 2.4.3. Show the following are equivalent for projections P and Q:

1. P and Q are disjoint as elements of a quantum logic.

2. The range of P and Q are orthogonal subspaces.

3. PQ = QP = 0.

Additionally, show that if P and Q are disjoint, then their least upper bound as elements in a quantum logicP ∪Q is given by their sum as projections P + Q.

2.5 States and Observables as Operators

In Section 2.4, we established the view that any quantum system S = (S ,O) can be determined bytaking the set of questions to be a set of projections for some corresponding separable Hilbert spaceH. In order to better understand this realisation of a quantum system, we explore both the set ofobservables O and the set of states S .

We first begin with a discussion of observables. Recall from Theorem 2.2.4 that we can identifythe set of observables with the set of question valued measures. Working with our assumption thatthe underlying set of questions corresponds to projections on some Hilbert spaceH, we will showthat you can identify the set of observables O with the set of valued measures. We take some timeto recall operator valued measures from operator theory.

Definition 2.5.1. A map Q : B→ B(H) from the set of Borel sets B to the set of bounded operatorson H is called an operator valued measure if for any x, y ∈ H and any countable disjoint collectionBi of Borel sets we have ⟨

Q(⋃

Bi

)x, y⟩= ∑

i〈Q(Bi)x, y〉.

Exercise 2.5.1. If Q is an operator valued measure then for each pair x, y ∈ H, the map E 7→ 〈Q(E)x, y〉is a complex valued Borel measure.

When an operator valued measure has a range consisting of projection operators then it is calleda projection valued measure. The next exercise helps us square this terminology with question valuedmeasures introduced in Section 2.1.

20

Exercise 2.5.2. Let S = (O,S) be a quantum system. Assume we can identify the set of questions Q ⊆ Owith the set of projections on a Hilbert space H. Show that any question valued measure in the sense ofDefinition 2.1.1 is an operator valued measure in the sense of Definition 2.5.1. Conversely, show that ifQ : B→ B(H) is an operator valued measure such that Q(E) is a projection for all Borel sets E, then Q is aquestion valued measure.

The above exercises tell us that we can view the set of observables as the set of projection valuedmeasures. In order to go one step further, we need invoke a version of the spectral theorem statedbelow.

Theorem 2.5.2 (Spectral theorem for self-adjoint operators). LetH be a separable Hilbert space. Thereis a one-to-one correspondence between self-adjoint operators A on H and projection valued measuresQ : B→ B(H).

It should be noted that this correspondence is not restricted to bounded operators. That is, itis possible that some observables will give rise to unbounded self-adjoint operators. In order tobetter understand the correspondence between observables and the corresponding operator, a solidunderstanding of the Borel functional calculus is needed. Since these technicalities are beyond thescope of this Thesis, we refer the reader Chapters 7 and 8 of [Hal13]. A careful reading of thesechapters should allow one to tackle the following exercise.

Exercise 2.5.3. Let Q : B→ B(H) be a question valued measure (and hence a projection valued measure)and let A be the corresponding self-adjoint operator given by Theorem 2.5.2. Show that Q is bounded inthe sense of Definition 2.4.3 if and only if A is a bounded operator. Show that the operator norm of Aagrees with the norm of Q. Finally, show that the definitions of spectrum and resolvent agree, i.e show thatσ(Q) = σ(A) and ρ(Q) = ρ(A).

We conclude our discussion of observables as operators at this point and proceed to our viewof states as density operators. In Theorem 2.2.4, we identify the set of states S with a full andstrongly convex set of abstract states. Clearly, the set of all abstract states will be full and stronglyconvex. Below we discuss how we can always considered this to be the canonical choice. To begin,consider abstract states that are constructed from unit vectors of the underlying Hilbert space. Seethe exercise below for the details.

Exercise 2.5.4. LetH be a separable Hilbert space and φ ∈ H be a unit vector. Then the map P 7→ 〈Pφ, φ〉defines an abstract state on the set of questions (i.e projections) in the sense of Definition 2.1.4.

Given a unit vector φ ∈ H, we use mφ to denote the corresponding abstract state, called a purestate, on the set of questions. Note the map φ 7→ mφ defines a map from unit vectors inH to abstractstates. This map will not be injective, in particular mφ = mψ if φ = cψ for |c| = 1. More generally,if we have a sequence of unit vectors φi ⊂ H, we can define an abstract state m on the set ofquestions by taking an infinite convex combination

m(P) = ∑i

timφi(P).

It can be shown that the set of all abstract states on the set of questions is a convex set and thepure states are extreme points of this convex set. Moreover, any abstract state can be written asa (possibly infinite) convex combination of pure states. Using this fact, we can give a sufficientcondition for the state space S in an quantum system to be taken to be the set of all abstract states.

Theorem 2.5.3. Suppose for each non-zero question Q ∈ Q, there exists a state α ∈ S such that mα(Q) = 1.Then the set of states S is in one-to-one correspondence with the set of abstract states.

21

Remark 2.5.4. Mackey proves this theorem after introducing an additional axiom. We insteadincluded it in the hypothesis.

Recall in Definition 2.4.5 we introduced the expected value mα corresponding to a state α. Inthe next exercise, we show how this expected value corresponds to the more common definitionin quantum information theory courses. (Again, a good understanding of the Borel functionalcalculus and the spectral theorem is required for the following exercise.)

Exercise 2.5.5. For any state α ∈ S that corresponds to a pure state mφ and for any observable A withcorresponding self-adjoint operator A′, we have

mα(A) = 〈A′φ, φ〉 = mφ(A′).

Now that we have established when it is reasonable to identify the set of states S with the set ofall abstract states, we proceed to identify states as density operators. In order to do this, we firstrecall the set of trace-class and density operators from operator theory.

Definition 2.5.5. LetH be a separable Hilbert space. An operator A ∈ B(H) is called trace-class iffor some (equivalently all) orthonormal basis en,

∑n〈(A∗A)1/2en, en〉 < ∞.

If A is a trace-class operator we define the trace Tr(A) by

Tr(A) = ∑n〈Aen, en〉.

Definition 2.5.6. A trace-class operator A that is positive semi-definite with Tr(A) = 1 is called adensity operator.

For the next three exercises, a decent background in basic functional analysis and operatortheory is required. For example, any course that covers content similar to the first 4 Chapters of[Ped89] should suffice.

Exercise 2.5.6. The set of trace-class operators is a two sided ideal in B(H).

Exercise 2.5.7. If A is a trace-class operator such that A ≥ 0 (i.e positive semi-definite and Tr(A) = 1)then we can define an abstract state on the set of projections by the map p 7→ Tr(AP).

Thus every density operator corresponds to an abstract state; this correspondence is actuallyone-to-one. Indeed, if mφ is a pure state, one can define the projection onto the one dimensional Pφ.Then Pφ is a density operator. Furthermore, for any projection P we have Tr(PφP) = 〈Pφ, φ〉 = mφ.We can do the same for mixed states, i.e. for states that are not pure.

Exercise 2.5.8. Let m = ∑i tiφi be a mixed abstract state on the set of projections. Then A = ∑i tiPφi is adensity operator with Tr(A) = 1. Furthermore, for all projections P we have the equality

m(P) = Tr(AP).

These exercises combined with the correspondence established between self-adjoint operatorsand observables allows us to conclude the following theorem.

22

Theorem 2.5.7 (Observables and States as operators). Let S = (S ,O) be a quantum system whose setof questions Q ⊆ O is taken to be the set of projections on a Hilbert spaceH. Then the set of observables Ois in one-to-one correspondence with the set of self-adjoint operators onH and, if we assume the hypothesisof Theorem 2.5.3, there is a one-to-one correspondence between the set of states S and the set of densityoperators.

We include one final exercise that requires the reader to have a good understanding of thecorrespondence above.

Exercise 2.5.9. Let B ∈ O be an observable with corresponding self-adjoint operator B′ ∈ B(H) and letα ∈ S be a state with corresponding density operator A. If mα is the expected value outlined in Definition2.4.5 we have the equality

mα(B) = Tr(AB′).

2.6 Quantum Strategies for non-local games

We now return to the topic of non-local games. Recall, in this setting two players, Alice andBob, collaborate to win a game played against a referee. The players are allowed to discuss andcoordinate an agreed upon strategy before the start of the game. Once the game starts they areforbidden to communicate with one another and each given a single question and each providea single response. The players win or lose this single round game based on the questions andanswers provided. Crucially, since the players do not communicate once the game commences theyare not aware of the question the player received.

Formally, a non-local game is described by a tuple (V, IA, IB, OA, OB, π) consisting of finite inputand outputs sets for Alice and Bob and a rule function V : IA × IB ×OA ×OB → 0, 1 as well asa distribution π on IA × IB. Alice and Bob are given inputs x and y respectively with probabilityπ(x, y) and produce respective outputs a and b. The players win when V(x, y, a, b) = 1. A strategyfor this game consists of a conditional probability function, or correlation, P = P(a, b|x, y), thatdetermines the probability Alice and Bob respond with answers a and b when given questions xand y. In order to model the fact that the players are separated we restrict the players to use onlyso called non-signalling strategies. That is, we require the conditional probabilities for Alice andBob PA(a|x), PB(b|y) to be well defined. The following sets of correlations outlined below are allexamples of non-signaling strategies.

Deterministic strategies are the natural way to model the strategies Alice and Bob can make ifthey are restricted by the physical rules of classical mechanics. In this scenario Alice and Bob planto pick a predetermined output for each possible input they receive.

Definition 2.6.1 (Deterministic Strategies). A strategy, P, for a non-local game (V, IA, IB, OA, OB, π)is called deterministic if there exist functions f : IA → OA and g : IB → OB such that

P(a, b|x, y) = δa( f (x))δb(g(y)).

Classical strategies are not restricted to only include deterministic strategies. More generally,Alice and Bob can employ randomness when determining their responses.

Definition 2.6.2 (Classical Strategies). A strategy, P, for a non-local game (V, IA, IB, OA, OB, π) iscalled classical if there exists a probability space (Ω, µ) such that for each x ∈ IA and y ∈ IB thereexist fx : Ω→ OA and gy : Ω→ OB with P(a, b|x, y) = µ( f−1

x (a)⋂

g−1y (b)).

23

An alternative set of strategies can be determined if Alice and Bob are allowed to take advantageof the axioms described in Section 2.1. Here we imagine that Alice and Bob will prepare a pure stateα and for each input x ∈ IA, that Alice receives she has a set of questions, Pa

x : a ∈ OA. Similarly,for each input y ∈ IB, Bob has a set of questions Qb

y : b ∈ OB. Given input x Alice will returnoutput a only if the answer to question Pa

x is yes. Likewise, Bob, will return output b exactly whenthe answer to question Qb

y is yes.We make two additional requirements on what are the possible questions Alice and Bob can

use. Firstly, in order for Alice to always make a reply we require that for any state α Alice alwaysreceives a yes response to one of her questions. The same requirement is made for the questionsBob receives. In the language our axioms this corresponds to the requirement that for every x, ywe have ∑a Pa

x = ∑y Qby = 1. Secondly, the non-signalling condition requires that conditional

probabilities, PA(a|x) and PB(b|y), are well-defined. This corresponds to the requirements that eachpair of questions Pa

x and Qby are simultaneously answerable, in the sense of Definition 2.3.1. Using

the correspondence outlined in Theorem 2.5.7 we know that the questions of Alice and Bob can beidentified with projections on some Hilbert space H and their shared pure state α can be identifiedwith a unit vector. Furthermore, the non-locality condition can be shown to be equivalent to arequirement that Alice’s projections commute with Bob’s. We then obtain the following definitionfor commuting operator strategies.

Definition 2.6.3 (Commuting Operator Strategies). P is a commuting operator strategy for a non-local game (V, IA, IB, OA, OB, π) if there exists a tuple (H, φ ∈ H, Pa

x, Qby), where H is a Hilbert

space, φ ∈ H is a unit vector, and Pax and Qb

y are families of projections that satisfy,

1. ∑a Pax = I for all x ∈ IA

2. ∑b Qby = I for all y ∈ IB

3. Pax Qb

y = QbyPa

x for all x, y, a, b

4. P(a, b|x, y) = 〈Pax Qb

yφ, φ〉.

One may consider the requirement that the questions of Alice and Bob are simultaneouslyanswerable too strong. Indeed, since Alice and Bob only ask questions with respect to a pre-determined state, α, it makes sense to consider strategies that Alice and Bob can employ when theycan consider questions that are only simultaneously answerable with respect to this chosen stateα. In the operator theory formalism this would correspond to a requirement Pa

x Qbyφ = Qb

yPax φ. We

refer to these strategies as state-commuting operator strategies.Alternatively, we can consider some sets of strategies that are formulated by placing stronger

requirements. The so called tensor product strategies are determined by requiring the Hilbert spaceH to have a decomposition in terms of tensor products as H = HA ⊗ HB. The projections forAlice and Bob are required to only act non-trivially on HA and HB, respectively. It was originallyconjectured by Tsirelson that the closure of the set of tensor product strategies was equal to the setof commuting operator strategies. This was recently shown to be incorrect [JNV+20].

In the case that the underlying Hilbert space H is restricted to be finite dimensional, the set ofcorresponding strategies is simply referred to as quantum strategies.

24

Chapter 3

Sums of Squares

Various natural notions of positivity appear throughout mathematics. In some cases, such as theChoi-Effros characterization of abstract operator systems, positivity really explains much of theunderlying structure [CE77]. One of the most intuitive notions of positivity is given by the squareof a number. In the case of the complex numbers C, it is straightforward to determine if a number αis positive, namely α is positive if we can express it as a hermitian square α = ββ. Informally wecan view the decomposition ββ here as a witness of the positivity of α.

Finding such a witness in various different contexts is the essence of sum of squares (SOS)techniques. The high level idea underlying this approach is that if an object satisfies some naturalnotion of positivity then there should be an obvious explanation, or SOS proof, of this fact. It is notalways the case that an SOS certificate is necessary for positivity.

The 17-th problem in Hilbert’s famous list of unsolved problems aimed to characterize when areal polynomial in several variables takes only non-negative values. In 1888, Hilbert [Hil88] provedthat not all such polynomials are sums of squares: the first explicit counter example was due toMotzkin [Mot67], with later counter examples due to Robinson [Rob69], Choi [CL76], and Lam[CL77]. It should be noted that for each of these counterexamples, the sum of the coefficients iszero. A proper characterization was famously given by Emil Artin in 1927, who confirms that suchpolynomials must be a sum of squares of rational functions [Art27] in any number of variables.

In this Chapter we are concerned with generalizations of these ideas to the setting of non-commutative algebras. In Section 3.4 we connect these concept to the topic of non-local games.

3.1 Non-commutative Sums of Squares

In 2002, a highly celebrated paper due to Helton [Hel02], similar considerations were made for socalled non-commutative polynomials. Informally, non-commutative polynomials are polynomialsin several variables, X1, . . . Xn, in which XiXj is not identified with XjXi. Instead of evaluating suchpolynomials at tuples of points they are evaluated at tuples of real matrices. Helton shows that if anon-commutative polynomial returns a positive semi-definite matrix under all evaluations then itmust be expressible as a sum of squares. A simplified argument due to McCollough and Putinarextends Helton’s result to the complex numbers [MP05].

A more formal description of these results can be stated in terms of ∗-algebras. Indeed, non-commutative polynomials can be viewed as elements in the the complex algebra A, that is freelygenerated by variables X1, . . . Xn. We can equip A with a formal involution by extending themap Xi 7→ X∗i in the natural way. Note, we do not require any relations to hold between Xi andX∗i . We call A, equipped with this involution, the free ∗-algebra in n variables. In this setting,

25

evaluating non-commutative polynomials at a tuple of matrices corresponds to a finite dimensional∗-representation of A. Using this formalism we can state the results of Putinar and McCollough asfollows.

Theorem 3.1.1. Let A be the free ∗-algebra in n variables. If a hermitian element f ∈ A is positive underall finite dimensional ∗-representations then there exists g1, . . . gm ∈ A such that f = ∑i gig∗i .

In Section 3.4 we explain the connections between non-commutative sums of squares andstrategies for non-local games. Theorem 3.1.1 is often invoked in the context of sums of squaresproofs for non-local games. As we will see in Section 3.4 the ∗-algebra A is not exactly the rightobject to work with in the context of non-local games. This is because the operators that Alice andBob use to make a quantum strategy for a non-local game satisfy some relations. For example, in aquantum strategy for the CHSH game, Alice and Bob both use observables of order 2 that commutewith one another. The correct language to describe these relations is found by extending the ideasums of squares to group rings.

3.2 Sums of Squares for Group Rings

Let C[G] be the ∗-algebra of a discrete group G. As in the case of non-commutative polynomials,we use ∗-representations to play the role of evaluation of our “polynomials”. In this setting ∗-representation correspond to unitary representations of the group. In order to obtain an analgue ofTheorem 3.1.1 for groups rings we need to understand the answer to the following question.

Question 3.2.1. If b ∈ C[G] is hermitian and π(g) is positive semidefinite for all unitary representationsπ, then is b necessarily expressible as a sum of hermitian squares?

Answers to this question have been provided in the positive for G = Fn, the free group onn generators, and G = Z2. The free group case has been credited to Schmüdgen, although itoriginally appears in the following paper by Netzer and Thom [NT13]. Netzer and Thom creditthe work of Scheiderer in [Sch06], [Sch99] to resolve this question in the positive for Z2 and in thenegative for Z3 respectively.

A slightly stronger result for Fn is due to Ozawa [Oza13]. Ozawa shows that if b ∈ C[Fn] ishermitian with supp(b) ⊆ EE−1 and b is positive semidefinite under all unitary representationsthen b is expressible as b = ∑ aia∗i where supp(ai) ⊆ E. Here supp(b) denotes the support of b.In [Rud63] Rudin shows that there exists an element f ∈ C[Z2] with support that is contained inthe set (i, j) : |i| ≤ 2N, |j| ≤ 2N for some integer N ≥ 3, satisfying χ( f ) ≥ 0 for all χ ∈ T2 butf can not be expressed as a sum b = ∑ aia∗i where for each i the support of ai is contained in theset (i, j) : 0 ≤ i ≤ N, 0 ≤ j ≤ N. As a consequence we know Z2 does not satisfy this strongerversion of the sum of squares property.

In [Oza13] it is shown that, for any discrete group G, if b is a hermitian element in C[G] that ispositive under all unitary representations then b must be in the so called archimedean closure ofthe cone sums of hermitian squares. That is, for all ε > 0 we have b + ε1 is expressible as a sum ofhermitian squares.

In this section we obtain a new sum of squares result for the group ring of Zk. We considerhermitian elements whose coefficient sum is non-zero. We show that if b is such an element then bis positive under all unitary representations if and only if it is expressible as a sum of hermitiansquares. In order to obtain this result we borrow heavily from the results of Netzer and Thomwho introduce generalized representations of a group in their proof that Fn satisfies the sum of squaresproperty [NT13]. We use amenability of Zk which provides an important approximation property

26

for the left regular representation of Zk. This allows us to apply the techniques of Netzer and Thomto elements b ∈ C[Zn] that are not in the augmentation ideal.

3.2.1 Notation

Let G be a discrete group and let C[G] denote the usual group algebra of G. Elements of C[G] areformal finite linear combinations of elements from G. A typical element is of the form b = ∑g αggwith all but finitely many αg equal to 0. The group algebra also comes equipped with the followinginvolution, (∑g αgg)∗ := ∑g αgg−1. Let C[G]h denote the set of elements b in group algebra thatsatisfy b∗ = b. We let ∑2 C[G] denote the set of all elements b ∈ C[G] that are expressible as a sumof hermitian squares. That is, there exists elements f1, . . . fm such that b = ∑i f ∗i fi.

We define the augmentation homomorphism ε : C[G]→ C by,

ε(∑g

αgg) = ∑g

αg.

The kernel of ε is known as the augmentation ideal of C[G],

ω(G) := Ker(ε) =

a = ∑

gαgg ∈ C[G] : ∑

gαg = 0

.

We will also make use of ultrafilters and ultraproducts. Given an ultrafilter ω, let Rω denotethe ultrapower of the reals. We let C denote the algebraically closed field Rω[i]. Note that theultrapower Rω can be equiped with a topology in a natural way, see [Ban77]. Thus it makes senseto consider limits of sequences of numbers in C.

We also recall the definition of doubly commuting operators.

Definition 3.2.2. Let A and B be linear operators on a Hilbert space. We say that A and B doublycommute if [A, B] = [A, B∗] = [A∗, B] = 0.

3.2.2 Generalized Representations

In [NT13] Theorem 3.11 the following powerful seperation theorem is given.

Theorem 3.2.3 (Netzer and Thom). Let G be a discrete group and let b be an element in the group ringthat is not expressible as a sum of hermitian squares. Then there exists, an ultrafilter ω and a C-linearfunctional φ : C[G]→ C = Rω[i], with φ(a∗) = φ(a) such that,

φ(b) < 0 and φ(a∗a) > 0 for all a ∈ C[G]\0.

One can apply the famed GNS construction on φ to obtain a C-vector space equipped witha positive definite sesquilinear form that takes values in C. The details of this construction arecontained in the proof of Theorem 3.3.1. For a good review of this construction see [Dav96]. Moregenerally if C = Rω for some ultrafilter ω we have the following generalized notion of a Hilbertspace.

Definition 3.2.4. A vector space H over C is called a generalized Hilbert space if it comes equippedwith a C-valued positive definite sesquilinear form 〈·, ·〉.

27

This sesquilinear form gives rise to a C-valued norm on H. Note in our definition that we donot require H to be complete with respect to this norm.

We will use L(H) to denote the set of linear operators on this space. Since Rω may fail to satisfythe least upper bound property it is not possible to define an operator norm here. Nevertheless wecan still use this sesquilinear form to determine the adjoint of a linear operator as well as a unitarylinear map, U ∈ B(H) such that U∗U = UU∗ = I. Using this adjoint we can view L(H) as a ∗algebra.

3.2.3 Approximation Properties of the Ring of Integers

Amenability of a discrete group is a well studied subject. For good introduction to the topicin a context relevant to this paper see [Oza13]. In the case of Zn, amenability of this groupallows us to conclude some strong approximation properties about the left regular representationλ : Zn → L(C[Zn]). In particular we will make use of the following well known construction.

Proposition 3.2.5. There exists a sequence of unit vectors vi ∈ C(G) such that |〈λ(s)vi, vi〉C(G) − 1| → 0for all s ∈ Zn.

Proof. Let Fi be a Følner sequence for Zn. We then define

vi =1

|Fi|12

χFi .

We then confirm the following calculations

‖λ(s)vi − vi‖C(G) = ‖1

|Fi|12

χsFi −1

|Fi|12

χFi‖C(G)

=|sFi4Fi||Fi|

→ 0.

Note this also tells us that 〈λ(s)vi, vi〉C(G) → 1.

|〈λ(s)vi, vi〉C(G) − 1| = |〈λ(s)vi, vi〉C(G) − 〈vi, vi〉C(G)|= |〈λ(s)vi − vi, vi〉C(G)|≤ ‖λ(s)vi − vi‖C(G) → 0

where the last inequality is obtained by applying Cauchy Schwarz inequality.

3.3 A Weak Sum Of Squares Property

Given a generalized Hilbert space H, an operator X ∈ L(H), and integer m ∈ Z we introduce thefollowing notation,

X(m) =

Xm, m ≥ 0X∗

−m, m ≤ 0

.

Using this notation we are now ready to state the following result.

28

Theorem 3.3.1. let b = ∑(n1,...nd)αn1,...nd(n1, . . . , nd) ∈ C[Zd]h with b not in the augmentation ideal,

ω(Zd), and b /∈ ∑2 C[Zd]. Then there exists a d-tuple, X = (X1, . . . , Xd), of doubly commuting contrac-tions on a finite dimensional Hilbert space K, and a vector x ∈ K such that,⟨

∑(n1,...nd)

α(n1,...,nd)X1(n1) . . . Xk(nd)x, x

⟩< 0.

Proof. In order to keep the notation concise the following proof is given for the case Z2. The moregeneral case can be seen to follow from the same argument.

Let b = ∑ α(m,n)(m, n) be a hermitian element of the group ring that is not expressible as asum of hermitian squares. Then by [NT13, Thm. 3.11], there exists an ultrapower R = Rω, and acompletely positive linear functional φ : C[Z2]→ C := R[i] with,

φ(a∗) = φ(a), and φ(b) < 0.

Let A = C⊗C C[Z2]. Then A is a ∗-algebra over C and we can extend φ to a complete positivemap on A in a canonical way. We next apply the GNS construction on φ. Define the the followingideal in A,

N := a ∈ A : φ(a∗a) = 0.Let H represent the quotient A/N. H comes equipped with the C-valued form,

〈a + N, c + N〉 = φ(c∗a).

We define a generalized representation π : C[Z2]→ L(H) by π( f )h = f · h. and let ξ := 1+ N ∈ H.Note 〈bξ, ξ〉 = φ(b) < 0.

We can now construct the generalized representation λ⊗ π : Z2 → L(C(Z2)⊗C H) defined by,

λ⊗ π(g)(a⊗ h) = λ(g)a⊗ π(g)h.

Here we are viewing C(Z2)⊗C H as a C−vector space, equipped with the sesquilinear form

〈a1 ⊗ h1, b1 ⊗ h2〉 = 〈a1, a2〉C[Z2] · 〈h1, h2〉H.

We also need to introduce the following generalized representation λ⊗ I : Z2 → L(C(Z2)⊗C[Z2]

H) defined by,λ⊗ I(g)(a⊗ h) = λ(g)a⊗ h.

By proposition 3.2.5 there exists a sequence of vectors vi ∈ C[Z2] satisfying 〈λ(s)vi, vi〉 → 1 forall s ∈ Z2. We then see that 〈λ(b)vi, vi〉 = ∑ αm,n〈λ((m, n))vi, vi〉 → ∑ αm,n = 1. (We may assumethat ∑ αm,n = 1. If this was not the case then we can simply normalize since b /∈ ω(Z2).) We thusget,

〈λ⊗ π(b))(vi ⊗ ξ), vi ⊗ ξ〉 = 〈π(b)ξ, ξ〉H · 〈λ(b)vi, vi〉= 〈π(b)ξ, ξ〉H ·

(∑ αm,n〈λ((m, n))vi, vi〉

)→ 〈π(b)ξ, ξ〉H < 0.

Thus we can fix large enough i such that x = vi ⊗ ξ satisfies,

〈λ⊗ π(b)x, x〉 < 0.

29

Recall in Proposition 3.2.5, vi was defined using a finite set Fi ⊆ Z2. That is we can write x as,

x =1

|Fi|12

∑(m,n)∈Fi

δ(m,n) ⊗ ξ.

Define a new vector x′ ∈ C[Z2]⊗ H as

1

|Fi|12

∑(m,n)∈Fi

δ(m,n) ⊗ π(m, n)ξ.

Using Fell’s absorption principle we can confirm that,

〈λ⊗ I(b)x′, x′〉 = 〈λ⊗ π(b)x, x〉 < 0.

From here we wish to define a finite dimensional subspace of K ⊆ C[Z2]⊗ H with the goal ofeventually applying the transfer principle. Let us define integers D1 and D2 by D1 := max|x|, |y| :(x, y) is in the support of b and D2 := max|m|, |n| : (m, n) ∈ Fi. Then consider the followingsubspace,

K := spanδ(a,b) ⊗ π(m, n)ξ : |a|, |b| ≤ D1 + D2, and |m|, |n| ≤ D2.

It is clear from the definition of K that we have both x′ and λ ⊗ I(b)x′ ∈ K. We next letA := λ ⊗ I(1, 0) and B := λ ⊗ I(0, 1). We now wish to show PAP and PBP doubly commute,where P is the orthogonal projection onto K. The fact that such an orthogonal projection existsfollows from applying the Gram-Schmidt process to find an orthonormal basis of K. We explore thedifferent cases. Suppose k = δ(a,b)⊗π((m,n)ξ ∈ K and we have |a + 1| and |b + 1| ≤ D1 + D2. Thenwe have Ak, Bk, ABk, BAk ∈ K and hence,

PAPPBPk = PAPBk = PABk = PBAk = PBPPAPk.

We now turn our attention to the other cases. Consider k = δ(a,b) ⊗ π(m, n)ξ ∈ K with |a + 1| >D1 + D2. Note that we will have Ak, ABk = BAk,∈ K⊥. Indeed if k′ = δ(a′,b′) ⊗ π(m′, n′)ξ ∈ Kthen a + 1 6= a′ so δ(a,b) ⊥ δ(a′,b′) and δ(a,b+1) ⊥ δ(a′,b′) in C[Z2] and so,

〈Ak, k′〉 = 〈δ(a+1,b), δ(a′,b′)〉 · 〈π(m, n)ξ, π(m′n′)ξ〉 = 0

〈ABK, k′〉 = 〈δ(a+1,b+1), δ(a′,b′)〉 · 〈π(m, n)ξ, π(m′n′)ξ〉 = 0.

Hence,PAPPBPk = PBPPAPk = 0.

The case for k = δ(a,b) ⊗ π(m, n)ξ ∈ K with |b + 1| > D1 + D2 is symmetric so we also get,

PAPPBPk = PBPPAPk = 0.

Let X = PAP and Y = PBP. Above we have shown that X commutes with Y and a similarargument will show that X commutes with Y∗ and Y commutes with X∗.

We also have the following identity,⟨∑

(m,n)∈supp(b)α(m,n)X(m)Y(n)x′, x′

⟩=⟨λ⊗ I(b)x′, x′

⟩< 0. (3.3.1)

30

Recall that, for m, n ∈ Z we have,

X(m) =

Xm, m ≥ 0X∗

−m, m ≤ 0

and Y(n) =

Yn, n ≥ 0Y∗−n

, n ≤ 0

To conclude proof we need to apply a version of the Lefschetz transfer principle. Let d denotethe dimension of finite dimensional vector space K over C. We can view X as a d × d matrixX = (xi,j) with entries from C and we can view k ∈ K as the tuple k = (k1, k2, . . . kd) ∈ Cd. SinceX = PAP we have for all k ∈ K we have ‖Xk‖ ≤ ‖k‖. A similar consideration can be made for Y.We then can conclude the following statements about C,

∃ X = xi,j ∈ Cd×d such that ∀(k1, . . . , kd) ∈ Cd, we have ∑j

∣∣∣∣∣∑ik jxi,j

∣∣∣∣∣2

≤∑j|k j|2.

∃ Y = yi,j ∈ Cd×d such that ∀(k1, . . . , kd) ∈ Cd, we have ∑j

∣∣∣∣∣∑ik jyi,j

∣∣∣∣∣2

≤∑j|k j|2.

∃x′ ∈ Cd such that 〈∑ αm,nX(m)Y(n)x′, x′〉 < 0.

[X, Y] = [X, Y∗] = [X∗, Y].

We can apply the transfer principle to these first order statements to conclude the following:There exists a finite dimensional Hilbert space K along with vector x ∈ K and doubly commut-

ing contractive operators X and Y on K such that the identity in (1) holds.

We will spend the rest of this section showing how one can use Theorem 3.3.1 to show Zk

satisfies a weak sum of squares property. We state the well known dilation results due to Sz-Nagyand Foias for convenience.

Theorem 3.3.2 (Sz-Nagy-Foias). Let Tini=1 be a family of doubly commuting contractions on a Hilbert

space K. Then there exists a Hilbert spaceH containing K as a subspace, and a family of doubly commutingunitaries Uin

i=1 onH, such that

T1(k1) . . . Tn(kn) = PKUk11 . . . Ukn

n |K.

Applying this dilation theorem to our previous result we get the following.

Corollary 3.3.3. Let k ∈N and b ∈ C[Zk]h with b /∈ ω(Zd). If π(b) ≥ 0 for all unitary representationsthen b ∈ Σ2C[Zd].

Proof. Given such an element, if b /∈ ∑2 C[Zd] then by Theorem 3.3.1 there exists a d-tuple, X =(X1, . . . , Xd), of doubly commuting contractions on a finite dimensional Hilbert space K,and avector x ∈ K such that, ⟨

∑(n1,...nd)

α(n1,...,nd)X1(n1) . . . Xd(nd)x, x

⟩< 0.

31

Applying Theorem 3.3.2 there exist doubly commuting unitary operators U1, . . . Ud on a HilbertspaceH ⊇ K such that, ⟨

∑(n1,...nd)

α(n1,...,nd)U1(n1) . . . Ud(nd)x, x

⟩< 0.

Since this tuple of unitaries doubly commute then they can induce a unitary representation π :Zd → B(H) defined by

π((0, 0, . . . , 1 . . . , 0))︸ ︷︷ ︸i

= Ui.

Finally since 〈π(b)x, x〉 < 0 we obtain a contradiction.

3.4 Non-local games and Sums of Squares

3.4.1 Groups and Commuting Operator Strategy Correspondence

In [PHMS19] the authors describe a correspondence between unitary representations of a particularfamily of groups and the operators of Alice and Bob in a commuting operator strategy. Below weoutline some of the details of this correspondence. Once this correspondence is established we candescribe the connection between the sum of squares property for these groups and the quantumvalue of a non-local game.

In Chapter 2 we saw the formal definition of commuting operator strategies for non-local games.For convenience we restate the definition here.

Definition 3.4.1. [Commuting Operator Strategies] P is a commuting operator strategy for a non-local game (V, IA, IB, OA, OB, π) if there exists a tuple (H, φ ∈ H, Pa

x, Qby), where H is a Hilbert

space, φ ∈ H is a unit vector, and Pax and Qb

y are families of projections that satisfy,

1. ∑a Pax = I for all x ∈ IA

2. ∑b Qby = I for all y ∈ IB

3. Pax Qb

y = QbyPa

x for all x, y, a, b

4. P(a, b|x, y) = 〈Pax Qb

yφ, φ〉.

For each input the measurement systems of Alice and Bob can be used to construct a unitaryoperator in a canonical way. Let n = |IA| and m = |OA|. For convenience we label the input andoutput set for Alice as IA = 0, . . . , n− 1 and OA = 0, 1, . . . , m− 1. If we let ω = e

2πim then for

each x ∈ IA we can define the following order m unitary operator,

Ux :=m−1

∑a=0

ωaPax .

Given any collection of n such unitaries Uxx∈IA the universal property of free-products allowsus to define a representation for the group F(n, m) := Zm ∗Zm ∗ · · · ∗Zm.

Similarly, if we take r = |IB| and s = |OB| and we let µ = e2πi

s then one can obtain a collectionof order r unitaries via the formula

Vy :=s−1

∑b=0

µbQby.

32

The collection Ux, Vy determines a unitary representation for the group F(n, m)×F(r, s).In [PHMS19] a converse to the above construction also holds. That is, given any unitary repre-

sentation of the group F(n, m)×F(s, r) one can construct a collection of commuting projectionsfor Alice and Bob that satisfy 1., 2. and 3. for Definition 3.4.1. To see this we let Ux, Vy de-note the canonical generators of the group and define the following elements in the group ringC[F(n, m)×F(s, r)],

Pax : =

1m

m−1

∑k=0

(ω−aUx)k. (3.4.1)

Qby : =

1s

s−1

∑k=0

(µ−bVy)k. (3.4.2)

If π : C[F(n, m)×F(s, r)]→ B(H) is a unitary representation then π(Pax ), π(Qb

y) will be such a setof projections for Alice and Bob.

3.4.2 Sum of Squares Certificates for Games

Given a non-local game G = (V, IA, IB, OA, OB, π) and strategy P = P(a, b|x, y) we can calculatethe probability of winning, wP, as

wP = ∑x,y,a,b

V(x, y, a, b)π(x, y)P(a, b|x, y).

The commuting operator value of the game G is then taken to be the supremum of wP over allcommuting operator strategies P. To each such game we can define a particular element of thegroup algebra C[F(n, m)×F(s, r)] called the game polynomial fG defined by,

fG := ∑x,y,a,b

V(x, y, a, b)π(x, y)Pax Qb

y.

Using the correspondence between commuting operator strategies and unitary representationsof the associated groups we obtain the following result.

Theorem 3.4.2. Given a non-local game (V, IA, IB, OA, OB, π), λ is an upper bound for the commutingoperator value of this game if and only if for all unitary representations, π of C[F(n, m)×F(s, r)], we haveπ(λ1− fG) is a positive semi-definite operator.

We know from [Oza13] that if a hermitian element in the group ring C[F(n, m)× F(s, r)] ispositive under all representations then it must be in the so called archimedian closure of theelements expressible as sums of hermitian squares. Consequently we have the following result.

Theorem 3.4.3. Given a non-local game G = (V, IA, IB, OA, OB, π), λ is a strict upper bound for thecommuting operator value of this game if and only if λ1− fG is expressible as a sum of hermitian squares.

In [NPA08] a useful semi-definite program is given to determine if λ is a strict upper bound onthe commuting operator value of a given non-local game. In [PNA10] the authors provide a semi-definite program to obtain a sum of squares certificate for strict upper bounds on the commutingoperating value. The answer to Question 3.2.1 is still not known for the group F(n, m)×F(s, r) andhence it is not known if the commuting operator value of a game will always have a correspondingsum of squares certificate.

33

Conjecture 3.4.4. If λ is the commuting operator value for for a non-local game, G, then λ1 − fG isexpressible as a sum of hermitian squares.

In [Oza13], a stronger sum of squares property is shown to hold for the free group. Morespecifically, the answer to the following question is yes for the case G is the free group with kgenerators.

Question 3.4.5. Suppose b ∈ C[G] is hermitian with the support of b contained in E−1E. If π(b) ispositive semidefinite for all unitary representations π, then is b necessarily expressible as a sum of hermitiansquares, b = ∑i f ∗i fi, with each fi having support in E?

By Tarski–Seidenberg theorem it is a decidable problem to determine if an element b ∈ C[G]is is expressible as b = ∑i f ∗i fi with each fi having support in E. In [Slo19] Slofstra shows thatdetermining whether or not a non-local game has commuting operator value strictly less then 1 isan undecidable problem. As a consequence of this fact we get the following result. This argumentwas shared with me in a private email exchange with William Slofstra.

Theorem 3.4.6. The answer to Question 3.4.5 is no for the case G = F(n, m)×F(s, r).

A non-local game G = (V, IA, IB, OA, OB, π) is called synchronous when IA = IB and OA = OBand v(x, x, a, b) = 0 whenever a 6= b. In the case of synchronous games the representation theorytheory of the group F(n, m) determines the possible commuting operator strategies for Alice andBob. It is still not known if the answer to either Question 3.2.1 or Question 3.4.5 is yes. The followingconjecture is then worth investigation.

Conjecture 3.4.7. If λ is the commuting operator value for for a synchronous non-local game then λ1−fG = ∑i f ∗i fi, with each fi having support in E, where E is a set that is computable using the support ofλ1− fG .

34

Chapter 4

Quantum Graphs

4.1 Introduction

Given a graph on n vertices one can associate two different subspaces of the n× n matrices thatencode all of the information of the graph. This has motivated the generalization of several wellknown graph theoretic concepts to a larger class of objects.

In [DSW13], Duan, Severini, and Winter describe a version of non-commutative graph theorywhose underlying objects consist of submatricial operator systems. The aforementioned authorsgeneralize the independence number and Lovász theta number to submatricial operator systems.

In [Sta16], Stahlke works with a similar but distinct definition of a non-commutative graph.Instead of working with submatricial operator systems, Stahlke associates a subspace of matriceswhose elements all have zero trace to a graph. Stahlke generalizes several classical graph theoryconcepts to these traceless subspaces including the chromatic number, clique number and notion ofgraph homomorphism.

Thus, there are two quite different subspaces of matrices to associate to graphs that lead totwo different ways to create a non-commutative graph theory. In this paper we discuss boththe submatricial operator system and submatricial traceless self-adjoint operator space definitions of anon-commutative graph.

There is currently no notion of the complement of a non-commutative graph that generalizesthe graph complement. By working with both of the above definitions we are able to generalize thecomplement of a graph using the orthogonal complement with respect to the Hilbert-Schmidt innerproduct. We conclude this section by reviewing the definition of several non-commutative graphparameters and show that some of these parameters can be approximated by evaluating classicalgraph parameters.

In [Lov79] Lovász introduced his well known theta number of a graph, θ(G). Lovász showsthat this number determines the following bounds on the independence number, α(G), and thechromatic number of the graph complement χ(G).

α(G) ≤ θ(G) ≤ χ(G).

These two inequalities are often referred to as the Lovász sandwich theorem. In [DSW13], it isshown that that the independence number of a submatricial operator system is bounded above byits Lovász number. This provides the first inequality for a generalized Lovász sandwich theorem.In [Sta16] Stahlke introduces a version of the chromactic number denoted χSt, that generalizes thesecond inequality.

35

In section 4.3 we introduce new generilzations of the chromatic number, χ0 and χ, that providelower and upper bounds on χSt. Using χ we provide a simplified proof of a weaker sandwichinequality. The advantage is that we can answer a question posed by Stahlke by generalizing theequation χ(G)ω(G) ≥ n to non-commutative graphs.

Given two graphs G and H the Cartesian product is the graph GH with vertex set V(G)×V(H) and edge relation given by (v, a) ∼ (w, b) if one of v ∼G w and a = b or v = w and a ∼H bholds. A Theorem of Sabidussi tell us χ(GH) = maxχ(G), χ(H) for any G and H. We introducea Cartesian product and establish a generalization of this result for submatricial traceless self-adjointoperator spaces in Section 4.4 . In section 4.4 we also establish a categorical product for submatricialtraceless self-adjoint operator space and extend a Theorem of Hedetniemi to submatricial tracelessself-adjoint operator spaces.

4.1.1 Notation

Let Mn denote the vector space of n× n matrices over C. This vector space can also be viewedas a Hilbert space using the inner product 〈A, B〉 := tr(B∗A). By a submatricial operator system,we mean a linear subspace S of Mn for which the identity matrix I belongs to S and for which Sis closed under the adjoint map ∗. A submatricial traceless self-adjoint operator space is a linearsubspace J ⊂ Mn for which J is closed under the adjoint operation ∗ and for which given anyA ∈ J , the trace of A is zero.

If S and T are two submatricial operator systems, then a linear map φ from S into T is calledcompletely positive (cp) if for each positive integer n, and for each positive semi-definite matrixX = [xi,j]i,j in Mn(S), the matrix

φ(n)(X) :=[φ(xij)

]i,j

in Mn(T) is positive semi-definite. We say that the cp map φ is unital and completely positive (ucp)if φ maps the identity I to the identity I. We say that φ is completely positive and trace preserving(cptp) if tr(φ(X)) = tr(X) for all X ∈ S. For more on cp maps see [Pau03].

We will also be using the following graph theory terminology. A graph G = (V, E) is an orderedpair consisting of a vertex set V and edge set E ⊂ V ×V. Since we are working with undirectedgraphs we require that if (i1, i2) ∈ E then (i2, i1) ∈ E. We say vertices i1 and i2 are adjacent, orconnected by an edge, and write i1 ∼ i2, whenever (i1, i2) ∈ E. An independent set of a graph G is asubset v ⊂ V such that for any two distinct elements i1, i2 ∈ v we have i1 6∼ i2. For a graph with nvertices it will be standard to consider the vertex set to be V = 1, . . . , n, which we will denote by[n].

4.2 Non-commutative graphs

A non-commutative graph is sometimes viewed as any submatricial operator system S. Non-commutative graphs have also been described as any submatricial traceless self-adjoint operatorspace J . In this section we review how one can view a classical graph as either of these objectswithout losing information about the graph itself. We also discuss several parameters for non-commutative graphs.

36

4.2.1 Non-commutative graphs as operator systems

Definition 4.2.1. Let G = (V, E) be a graph with vertex set [n]. Define SG ⊂ Mn by

SG := spanEi,j : (i, j) ∈ E or i = j.

Observe that for any graph G, SG will be a submatricial operator system. In [OP15], it is shownthat graphs G and H are isomorphic if and only if SG and SH are isomorphic in the category ofoperator systems. We discuss this in more details in 4.2.2.

Given a graph G, if vertices i, j are not adjacent, then eie∗j = Ei,j is orthogonal to the submatricialoperator system SG. Similarly if i1, . . . , ik is an independent set of vertices in G then for any j 6= kwe have eij e

∗ik

is orthogonal to SG. If v = (v1, . . . , vk) is an orthonormal collection of vectors in Cn

then v called an independent set for a submatricial operator system S ⊂ Mn if for any i 6= j, viv∗j isorthogonal to S.

Definition 4.2.2. Let S be a submatricial operator system. We define the independence number, α(S),to be the largest k ∈N such that there exists an independent set for S of size k.

A graph G = (V, E) has a k-colouring if and only if there exists a partition of V into k in-dependent sets. In [HPP16] Paulsen defines a natural generalization of the chromatic numberto non-commutative graphs. We say a submatricial operator system S ⊂ Mn has k-colouring ifthere exists an orthonormal basis for Cn, v = (v1, . . . , vn), such that v can be partitioned into kindependent sets for S.

Definition 4.2.3. Let S ⊂ Mn be a submatricial operator system. The chromatic number, χ(S), is theleast k ∈N such that S has a k-colouring.

For any submatricial operator system S ⊂ Mn we have χ(S) ≤ n since you can partition anybasis of Cn into n independent sets. In Theorem 4.2.12 we show that both of the above parametersprovide a generalization of the classical graph theory parameters, that is we show α(SG) = α(G)and χ(SG) = χ(G). This first equality is originally found in [DSW13] and the second can be foundin [HPP16].

Example 4.2.4. Consider the submatricial operator system S := spanI, Ei,j : i 6= j ⊂ Mn. Let u1, u2be two orthonormal vectors and let i be an element of the support of u1. Since u∗1u2 = 0 theremust be an element j 6= i of the support of u2. Then 〈u1u∗2 , Ei,j〉 = u1(i)u2(j) 6= 0. Thus we see thatα(S) = 1. This also tell us that χ(S) = n.

As in [DSW13], given a graph G one can compute the Lovász theta number θ(G) as,

θ(G) = max‖I + T‖ : I + T ≥ 0, Ti,j = 0 for i ∼ j.

Here the supremum is taken over all n× n matrices and I + T ≥ 0 indicates that I + T is positivesemidefinite.

The following inequality is due to Lovász. For a good self-contained review please see [Knu94].

Theorem 4.2.5. Let G be a graph and G be the graph complement of G. Then,

α(G) ≤ θ(G) ≤ χ(G).

In order to obtain an generalization of 4.2.5 we need to identify the the appropriate generaliza-tion of a graph complement. Given a submatricial operator system S ⊂ Mn we use the orthogonalcomplement S⊥ to generalize the graph complement. Note that the orthogonal complement of a

37

submatricial operator system is no longer a submatricial operator system since it will fail to containthe identity operator. In fact since I ∈ S we will have tr(A) = 〈A, I〉 = 0 for every A element of S⊥.In [Sta16], Stahlke works with precisely these objects. We show that it is useful to consider bothsubmatricial operator systems and submatricial traceless self-adjoint operator spaces to generalizethe graph complement.

4.2.2 The complement of a non-commutative graph

In this section, we introduce the analogue of the notion of a graph complement for non-commutativegraphs. Using this, we define a notion of clique number independence number and chromaticnumber.

Definition 4.2.6. Let G be a finite graph with vertex set [n]. The traceless self-adjoint operator spaceassociated to G is the linear space

JG := spanEi,j : i ∼ j ⊂ Mn .

A traceless non-commutative graph is any submatricial traceless self-adjoint operator space.

Remark 1. The traceless self-adjoint operator space JG is the traceless non-commutative graph SG givenin [Sta16]. Given a finite graph G with vertex set [n], we have the identity J ⊥G = SG. This identityin particular suggests that the graph complement of a non-commutative graph should be its orthogonalcomplement. In [Sta16], Stalhke suggests that the graph complement of JG should be (JG +CI)⊥. However,this notion of complement would mean that JG 6= (JG + CI)⊥ for any graph with at least two vertices.We shall see that, so long as one is willing to pay the price of working with two different notions of anon-commutative graph, the orthogonal complement is the correct analogue of the graph complement.

Proposition 4.2.7. The traceless self-adjoint operator subspaces of Mn are exactly the orthogonal comple-ments of submatricial operator systems. That is, S is a submatricial operator system if and only if S⊥ is atraceless self-adjoint operator space.

Proof. If S is an operator subsystem of Mn then for any X ∈ S⊥, tr(X) = 〈X, I〉 = 0. As well, ifX ∈ S⊥, for any Y ∈ S, tr(XY) = tr(Y∗X∗) = 0. This proves that S⊥ is a traceless self-adjointoperator space. Conversely, if S is a traceless self-adjoint operator space, then S⊥ contains I sincefor all X ∈ S, 〈X, I〉 = tr(X∗ I) = 0. If X ∈ S⊥ then 〈X∗, Y〉 = tr(XY) = tr(Y∗X∗) = 0. Therefore,X∗ ∈ S⊥. This proves that S⊥ is an operator system.

Proposition 4.2.8. If G is a graph with vertex set [n] then S⊥G = JG.

Proof. Observe that for i, j, k, l ∈ [n], Eij ∈ S⊥G if and only if for all k 'G l, tr(EijEkl) = 0. This is onlypossible if i ∼G j.

It is a result of Paulsen and Ortiz [OP15, Proposition 3.1] that two graphs G and H of the samevertex set [n] are isomorphic if and only if there is a n× n unitary matrix U for which USGU∗ = SH .

Corollary 4.2.9. Suppose that G and H are graphs with vertex set [n]. The graphs G and H are isomorphicif and only if there is an n× n unitary matrix U such that UJGU∗ = JH.

Proof. For any n× n unitary matrix U, (USGU∗)⊥ = UJGU∗. Since G and H are isomorphic if andonly if their graph complement is, the result follows.

38

Remark 2. In [Wea17] a qunatum graph is defined as a reflexive, symmetric quantum relation on a∗-subalgebraM⊆ Mn. In this framework a submatricial operator system S is indeed quantum graph whentakingM = Mn. This approach fails to provide a complement for a quantum graph since S⊥ will fail to be areflexive quantum relation on anyM⊆ Mn and hence will not be a quantum graph.

The notion of an independent set for an submatricial operator system was described solely interms of an orthogonality relation. We can similarly say that an orthonormal collection of vectorsv = (v1, . . . , vn) in Cn is an independent set for a submatricial traceless self-adjoint operator spaceJ ⊂ Mn if for any i 6= j, viv∗j is orthogonal to J . We say J has a k-coloring if there exists anorthonormal basis v = (v1, . . . , vn) of Cn, that can be partitioned into k independent sets for J .

Definition 4.2.10. Let J ⊂ Mn be a submatricial traceless self-adjoint operator space.

1. The independence number, α(J ), is the largest k ∈N such that there exists an independentset of size k for J .

2. The chromatic number χ(J ) is the least integer k such that J has k-colouring.

It is not hard to show that χ is monotonic and α is reverse monotonic under inclusion. This holdswhen considering these as parameters on submatricial operator systems as well as submatricialtraceless self-adjoint operator spaces.

Next we show that if G is a graph, SG and JG have the same independence number andchromatic number. We start with a lemma. The following proof is in [?, Lemma 7.28]:

Lemma 4.2.11. Let v1, . . . , vn be a basis for Cn. There exists a permutation σ on [n] so that for each i, theσ(i)th component of vi is non-zero.

Proof. Let A = [ai,j] denote the matrix with column i equal to vi. Since we have a basis, det(A) 6= 0.But

det(A) = ∑σ∈Sym([n])

sgn(σ)a1,σ(1) · · · an,σ(n) .

There must therefore be some σ for which the product a1,σ(1) · · · an,σ(n) is non-zero. This permutationworks.

It has been shown that α(G) = α(SG) and χ(G) = χ(SG) in [DSW13] and [HPP16] respectively.We are able to obtain the analogous results for submatricial self-adjoint operator spaces.

Theorem 4.2.12. Let G be a graph on n vertices, we have α(G) = α(SG) = α(JG) and χ(G) = χ(SG) =χ(JG).

Proof. The inclusion α(SG) ≤ α(JG) follow from reverse monotonicity. If i1, . . . , ik are an inde-pendent set of vertices in the graph G then we have that the standard vectors ei1 , . . . , eik is anindependent set for SG so we get

α(G) ≤ α(SG) ≤ α(JG) .

Next suppose that v1, . . . , vk are an independent set for JG. Then since v1, . . . , vk is an linearlyindependent set of vectors we can find a permutaiton σ on [n] so that

⟨vi, eσ(i)

⟩is non-zero for all i.

We note that if vertices σ(j) and σ(k) are adjacent in G then we have Eσ(j),σ(k) ∈ JG. But then

〈vjv∗k , Eσ(j),σ(k)〉 =⟨vj, eσ(j)

⟩ ⟨eσ(k), vk

⟩6= 0 a contradiction. Thus σ(1), . . . , σ(k) are an independent

set for the graph G so α(JG) ≤ α(G). The proof for χ follows the same argument.

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Recall for a classical graph G the clique number, ω(G), satisfies that ω(G) = α(G).

Definition 4.2.13. Let S be a submatricial operator system and let J be a submatricial tracelessself-adjoint operator space.

1. Define the clique number, ω(S), to be the independence number of the submatricial tracelessself-adjoint operator space S⊥.

2. Define the clique number, ω(J ), to be the independence number of the submatricial operatorsystem J ⊥.

It should be noted that the above definition ω(J ) of a traceless submatricial operator space isfirst mentioned in [Sta16]. We can use Theorem 4.2.12 to conclude that for any graph G we haveω(G) = ω(SG) = ω(JG).

The next proposition shows that α, ω, and χ may be computed purely from the associatedparameters for graphs. We can achieve this by associating a family of graphs to each submatricialtraceless self-adjoint operator space or submatricial operator system.

Definition 4.2.14. Given a submatricial operator system S ⊆ Mn and an orthonormal basis v =(v1, . . . , vn) we can construct two different graphs.

1. The confusability graph of v, with respect to S, denoted Hv(S), is the graph on n vertices withi ∼ j if and only if viv∗j ∈ S.

2. The distinguishability graph of v, with respect to S, denoted Gv(S) is the graph on n verticeswith i ∼ j if and only if viv∗j ⊥ S.

We can also define the confusability and distinguishability graphs of an orthonormal basisv = (v1, . . . , vn) with respect to submatricial traceless self-adjoint operator spaces J ⊆ Mn in thesame way. We would then have for S = J ⊥, Gv(S) = Hv(J ) and Hv(S) = Gv(J ). When it is clearwhat the underlying system or submatricial traceless self-adjoint operator space is we simply writeGv and Hv.

Theorem 4.2.15. Let J be a submatricial traceless self-adjoint operator space in Mn and let B denote theset of ordered orthonormal bases for Cn. We have the identities

α(J ) = supv∈B

α(Gv) ,

χ(J ) = infv∈B

χ(Gv) , and

ω(J ) = supv∈B

ω(Hv) .

The same identity holds if we replace J with a submatricial operator system in Mn.

Proof. Suppose v1, . . . , vc is a maximal independent set for J , that is for i 6= j we have viv∗j ⊥ J .We can extend this collection to an orthonormal basis v = (v1, . . . , vc, vc+1, . . . vn). Note that thevertices 1, . . . , c in the graph Gv are an independence set since for distinct i, j ∈ [c] we have viv∗j ⊥ J .This gives i ∼ j in Gv. Thus there is no edge between i and j in Gv. Therefore α(Gv) ≥ c so we haveα(J ) ≤ supv∈B α(Gv). Conversely, for each v ∈ B if i1, . . . , ic are an independent set for Gv thenij ∼ ik in GV . We then have vi1 , . . . , vic is an independent set for J . This gives α(J ) ≥ α(Gv).

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The proof of the second identity is similar. If χ(J ) = c then there exists orthonormal basisv = (v1, . . . , vn) and a partition P1, . . . Pc of [n] such that viv∗j ⊥ J for distinct i and j in the samepartition. Define a colouring f of Gv by having f (i) = l if and only if i ∈ Pl . We see that for i 6= j ifwe have f (i) = f (j) then viv∗j ⊥ J giving that i ∼ j in Gv so f is indeed a c colouring of Gv. Thisgives χ(J ) ≥ infv∈B χ(Gv). Conversely, if f is any c colouring of Gv for some v ∈ B then we canobtain a c colouring of J by partitioning [n] into sets P1, . . . Pc where i ∈ Pl if and only if f (i) = l.Then if distinct i, j ∈ Pl we have i ∼ j in Gv so viv∗j ⊥ J .

Lastly, suppose (v1, . . . , vk) is a collection of orthonormal vectors such that for distinc i, j wehave viv∗j ∈ J . We can extend this set to a orthonormal basis v = (v1, . . . vn) and we immediatelyget that the vertices 1, . . . , k form a clique in Hv. For the other direction note for any basisv = (v1, . . . vn) Hv has a clique i1, . . . , ik then v1, . . . vk will satisfy viv∗j ∈ J for distinct i and j.

We next extend the definition of Lovász’ theta function, [Lov79], to non-commutative graphs.This was first extended to submatricial operator spaces in [DSW13]. We introduce a naturalextension to submatricial operator systems as well.

Definition 4.2.16. Let S be a submatricial operator system and J be a submatricial traceless self-adjoint operator space. Define the theta number of a submatricial operator system, θ(S) and thecomplementary theta number of a submatricial traceless self-adjoint operator space, θ as follows.

1. θ(S) = sup‖I + T‖ : T ∈ Mn, I + T ≥ 0, T ⊥ S.

2. θ(J ) = sup‖I + T‖ : T ∈ Mn, I + T ≥ 0, T ∈ J .

Observe that θ(SG) = θ(G) and θ(JG) = θ(G) for all graphs G.

Example 3. Recall the previously mentioned submatricial operator system S := spanI, Ei,j : i 6= j ⊂ Mn.We see that θ(S) = n since we can take T to be the diagonal matrix with n− 1 for the 1, 1 entry and −1for all other diagonal entries. We also see that if v = (e1, . . . , en) is the standard basis for Cn then v is aclqiue for S and thus we have χ(S⊥) = 1. This shows that using the definition of the chromatic numberfrom [HPP16] we can not hope to generalize the Lovász sandwich theorem.

4.3 Non-commutative Lovász inequality

We see by the previous example that one needs a different generalization of the chromatic numberin order to obtain a Lovász sandwich Theorem for non-commutative graphs. Here we intro-duce the strong and minimal chromatic number of a submatricial operator system and provide ageneralization on Lovász theorem.

4.3.1 The Strong chromatic number

Let J ⊂ Mn be a submatricial traceless self-adjoint operator space. A collection of orthonormalvectors v = (v1, . . . , vk) in Cn is called a strong independent set for J if for any i, j,we have viv∗jis orthogonal to J . We say that J has a strong k-colouring if there exist an orthonormal basisv = (v1, . . . , vn) of Cn that can be partitioned into k strong independent sets for J . We will show inCorollary 4.3.6, χ(JG) agrees with the chromatic number of G, for any graph G.

Definition 4.3.1. Let J ⊂ Mn be a submatricial traceless self-adjoint operator space. The strongchromatic number, χ(J ), is the least k ∈N such that J has a strong k-colouring. If J has no strong-kcolouring then we say χ(J ) = ∞.

41

As with χ we have χ is monotonic with respect to inclusion.

Example 4. Suppose that S = C1 + spanEi,j : i 6= j ⊂ Mn and ζ is a nth root of unity. Definevk = (1, ζk, ζ2k, . . . , ζ(n−1)k). Observe that the vi are orthogonal and that vkv∗k belongs to S for all k. ThusS⊥ does have a strong-n colouring and we get χ(S⊥) ≤ n.

Example 5. Consider the submatricial traceless self-adjoint operator space J = C∆ ⊂ Mn where ∆ =diag(n− 1,−1,−1, . . . ,−1). Observe that J ⊂ S⊥. By monotonicity, χ(J ) ≤ χ(S⊥) ≤ n. It is knownthat θ(J ) = n (see [OP15, Remark 4.3]). We show in Theorem 4.3.7 that χ is bounded below by θ Thus wehave χ(J ) = χ(S⊥) = n.

In [Sta16] Stahlke introduces a different chromatic number for submatricial traceless self-adjointoperator spaces.

Definition 4.3.2. Let J and K be submatricial traceless self-adjoint operator spaces in Mn and Mmrespectively. We say that there is a graph homomorphism from J to K, denoted J → K, if thereis a cptp (completely positive and trace preserving) map E : Mn → Mm with associated Krausoperators E1, . . . , Er for which EiJ E∗j ⊂ K for any i and j.

Stalhke’s chromatic number of a submatricial traceless self-adjoint operator space J , denotedχSt(J ), is the least integer c for which there is a graph homomorphism J → JKc if one exists. Weset χSt(J ) = ∞ otherwise.

Observe that χSt is monotonic under graph homomorphism by construction.

Theorem 4.3.3. For any submatricial traceless self-adjoint operator space J ⊂ Mn we have χ(J ) ≥χSt(J ).

Proof. Suppose χ(J ) = r. There exists a orthonormal basis v1, . . . , vn that can be partitioned intostrong independent sets P1, . . . , Pr. By reordering the vectors, we may assume that whenevervi ∈ P` and vj ∈ P`+1, that i < j. By conjugating by the unitary U : vi 7→ ei, we get the inclusion⊕r

i=1 M|Pi | ⊂ UJ ⊥U∗ = (UJU∗)⊥.This then gives us (

⊕ri=1 M|Pi |)

⊥ ⊃ (UJU∗). We have that J → UJU∗ by conjugating by theunitary U. Similarly we have UJU∗ → (⊕r

i=1M|Pi |)⊥ by inclusion. Since χSt is monotonic with

respect to homomorphisms we get χSt(J ) ≤ χSt((⊕ri=1M|Pi |)

⊥) = χ(G) = r where G is the disjointunion of r complete graphs.

Corollary 4.3.4. If J ⊂ Mn is a submatricial traceless self-adjoint operator space for which for some basisv = (v1, . . . , vn), the diagonals viv∗i are orthogonal to J , then χSt(J ) ≤ n.

In [LPT17] it was shown that α(S) = α(Md(S)) for all submatricial operator systems S. Theproof they give will also work to show α(J ) = α(Md(J )) and χ(J ) = χ(Md(J )) for submatricialtraceless self-adjoint operator spaces J and d ∈N.

Recall that for d, n ≥ 1, the partial trace map is

Md ⊗Mn → Mn : X⊗Y 7→ tr(X)Y .

As is the case with χ we can approximate χ using the chromatic number for classical graphs.

Theorem 4.3.5. Let J ⊂ Mn be a submatricial traceless self-adjoint operator space. Suppose that BJdenotes the set of ordered orthonormal bases v = (v1, . . . , vn) of Cn for which viv∗i ∈ J for all i. For each

42

v = (v1, . . . , vn) in BJ , define the graph Gv with vertices [n] and edge relation given by i ∼ j if viv∗j isorthogonal to J . Then,

χ(J ) = infv∈B

χ(Gv) ,

whenever χ(J ) is finite.

Proof. The proof is exactly as in Theorem 4.2.15.

Corollary 4.3.6. For any finite graph G, χ(JG) = χ(G).

Proof. By Theorem 4.3.5, χ(JG) ≤ χ(Gv) where v = (e1, . . . , en). The complement of the graphGv is the graph G. This gets us the bound χ(JG) ≤ χ(G). As well, by Theorem 4.3.3, χ(G) =χSt(JG) ≤ χ(JG).

Using the strong chromatic number we are easily able to generalize other graph inequalitiesthat for now remain unanswered for χSt. In [Sta16] Stahlke asks if for all submatricial tracelessself-adjoint operator spaces J ⊂ Mn, one can show χSt(J )ω(J c) ≥ n, where J c is the pro-posed complement J c = (J + CI)⊥. The question is motivated by the simple graph inequalityχ(G)ω(G) ≥ n. Indeed for J ⊂ Mn a submatricial traceless self-adjoint operator space if wesuppose χ(J ) = k then we can find an orthonormal basis v = (v1, . . . , vn) and a partition of v intoindependent sets P1, . . . , Pk. By definition of ω(J ⊥) we know that |Pi| ≤ ω(J ⊥) for i = 1, . . . , k.Thus we have n = ∑i |Pi| ≤ ∑i ω(J ⊥) = χ(J )ω(J ⊥).

Using [Sta16], one can establish that θ(J ) ≤ χSt(J ) for any submatricial traceless self-adjointoperator space J ⊂ Mn: if c = χSt(J ), then there is a graph homomorphism J → JKc . In [Sta16,Theorem 19], it is shown that θn is monotonic under graph homomorphisms. We therefore get theinequality

θn(J ) ≤ θn(JKc) = θ(Kc) ≤ χ(Kc) = c .

We can now establish a Lovász sandwich Theorem for χ.

Theorem 4.3.7. Let S be a submatricial operator system. For any d ≥ 1, we have the inequalities

α(S) ≤ θ(S) ≤ χ(S⊥) .

Proof. The inequality α(S) ≤ θ(S) is a result in [DSW13, Lemma 7] so we will only prove theother inequality. Let v = (v1, . . . , vn) be an orthonormal basis that can be partitioned into k strongindependent sets for S⊥. Then consider the graph Gv(S⊥) as defined in Theorem 4.2.15. We haveχ(S⊥) = χ(Gv). There exists a unitary U ∈ Mn such that we get the the inclusion S ⊃ USGvU∗.Since θ is reverse monotonic under inclusion and invariant under conjugation by a unitary, weestablish the inequalities

θ(S) ≤ θ(SGv) = θ(Gv) ≤ χ(Gv) = χ(S⊥).

Similarly we get the follow inequality for any submatricial traceless self-adjoint operator spaceJ .

α(J ⊥) ≤ θ(J ) ≤ χ(J ).

It should be pointed out that using χSt(J ) ≤ χ(J ), as shown in Theorem 4.3.3 , and the factthat ω(J ) = α(J ⊥), one can obtain the the above inequality as a corollary of Corollary 20 in[Sta16] . In this sense the above can be considered as a simplified proof of a weaker result.

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4.3.2 The minimal chromatic number

In this section, we wish to construct a concrete definition of a homomorphism monotone chromaticnumber.

Definition 4.3.8. Let J ⊂ Mn be a submatricial traceless self-adjoint operator space. Define theminimal chromatic number of J , denoted χ0(J ), to be the least integer c for which there existsa basis v1, . . . , vn of Cn and a partition P1, . . . , Pc of [n] for which whenever i, j ∈ Ps, we have therelation viv∗j ⊥ J .

We note thatχ0 differs from χ since we no longer require that we are working with an orthonor-mal basis. This parameter agrees with the chromatic number for graphs. We define the competelybounded version of this parameter by χ0,cb(J ) = infd χ0(Md(J )).

Proposition 4.3.9. Let G be a finite graph. We have the relation χ(G) = χ0(JG).

Proof. Since χ0(JG) ≤ χ(G), it suffices to show that χ(G) ≤ χ0(JG). For this proof, let c beminimal and let v1, . . . , vn be a basis in Cn for which there is a partition P1, . . . , Pc of [n] such thatwhenever i, j in Ps, viv∗j ⊥ JG. We then have a permutation σ of [n] for which

⟨vi, eσ(i)

⟩is non-zero.

By conjugating JG by the permutation matrix defined by σ, assume that σ(i) = i for all i. Definethe c-colouring f : V(G) → [c] By f (i) = s for s such that i ∈ Ps. To see that this is a colouring,suppose not. There are then i ∼ j for which i, j ∈ Ps for some s. By definition then we have, Ei,jbelongs to JG. We observe then,⟨

viv∗j , Ei,j

⟩= tr(vjv∗i eie∗j ) = 〈vi, ei〉

⟨vj, ej

⟩6= 0 .

This is contradicts the fact that viv∗j ∈ J ⊥.

We recall the following result, which arises as a consequence of the Stinespring dilation Theorem(see [Sta16, Definition 7]).

Lemma 4.3.10. Let J ⊂ Mn and K ⊂ Mm be submatricial traceless self-adjoint operator spaces. Thereis a graph homomorphism J → K if and only if there is a d ≥ 1 and an isometry E : Cn → Cm ⊗Cd forwhich EJ E∗ ⊂ Md(K).

We use this equivalent characterization to show that χ0,cb is monotonic under graph homomor-phisms.

Theorem 4.3.11. Let J ⊂ Mn and K ⊂ Mm be submatricial traceless self-adjoint operator spaces. If thereis a graph homomorphism φ : J → K with d associated Kraus operators, then we have the inequality

χ0(J ) ≤ χ0(Md ⊗K) .

In particular, χ0,cb(J ) ≤ χ0,cb(K).

Proof. Suppose that P1, . . . , Pc is a partition of the set [d]× [m] and (wi : i ∈ [d]× [m]) is a basisfor which whenever i, j are in the same Ps, then wiw∗j ⊥ Md ⊗ K. By lemma 4.3.10, there is anisometry E for which the map φ : Mn → Md ⊗Mm : X 7→ EXE∗ sends J to Md ⊗K. Consider theset E∗wi : i ∈ [d×m]. This set spans Cn. To see this, for any v ∈ Cn, since Ev ∈ Cd ⊗Cm, thereare some λi for which Ev = ∑i λiwi. Multiplying on the left by E∗ tell us that v is spanned by theE∗wi. If i, j belong to the same Ps, then for any X ∈ J ,⟨

E∗wi(E∗wj)∗, X

⟩=⟨

wiw∗j , EXE∗⟩= 0 .

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For each i ∈ [c], let Ci = E∗wj : j ∈ Pi. We will define a sequence of linear subspaces V1, . . . , Vcfor which ∑c

i=1 Vi = Cn inductively. For the base case, set V1 = span C1. For i > 1, let

Vi = span

v ∈ span Ci : v 6∈ ∑

k<iVk

.

By construction, for distinct i and j, the vectors the Vi are linearly independent in relation tothe vectors of Vj and ∑i Vi = Cn. For each s, let Qs = vs,1, . . . , vs,ds be a basis in Vs, whereds = dim(Vs). Since each vector in Qs is a linear combination of the vectors in Cs, we get thatwhenever, i, j ∈ [ds], given any X ∈ J , ⟨

vs,iv∗s,j, X⟩= 0 .

The vectors vs,i : s ∈ [c], i ∈ [ds] then form a basis for Cn and are partitioned by the setsQs : s ∈ [c]. This proves that χ0(J ) ≤ χ0(Md ⊗K). If r ≥ 1 and E is an isometry for which themap φ : Mn → Md ⊗Mm : X 7→ EXE∗ sends J to Md(K), then the map

1⊗ φ : Mr ⊗Mn → Mr+d ⊗Mm : X⊗Y 7→ X⊗ φ(Y)

is a map implemented by conjugation by the isometry 1⊗ E. By lemma 4.3.10, 1⊗ E is a graph homo-morphism Mr(J )→ Mr(K). By the above proof, we get the bound χ0(Mr(J )) ≤ χ0(Mr+d(K)) ≤χ0,cb(K) for every r ≥ 1. This establishes the inequality

χ0,cb(J ) ≤ χ0,cb(K) .

Corollary 4.3.12. Let J be a submatricial traceless self-adjoint operator space. We have the inequality

χ0,cb(J ) ≤ χSt(J ) .

Proof. We first show that χ0,d(JG) = χ(JG) for any d ≥ 1 and any graph G. Let G[d] denote thegraph on vertices V(G)× [d] for which (v, i) ∼ (w, j) if v ∼ w in G. The projection G[d] → G :(v, i) 7→ v and the inclusion G → G[d] : v 7→ (v, 1) are graph homomorphisms. We therefore getby monotonicity of χ that χ(G) = χ(G[d]). On the other hand, we know that χ0(JG) = χ(G) =χ(G[d]) = χ0(Md(JG)) = χ0,d(JG). In particular, for any c ≥ 1, χ0,cb(JKc) = χ(Kc) = c.

Remark 6. We were unable to determine if χ0,cb = χ0. Never the less the obtain value by working withχ0,cb since we know it is a homomorphism monotone parameter.

4.4 Sabidussi’s Theorem and Hedetniemi’s conjecture

As an application of our new graph parameters, in this section, we generalize two results forchromatic numbers on graph products. For convenience we will let χ(X) = χ(X⊥) for X asubmatricial traceless self-adjoint operator space or a submatricial operator system.

Definition 4.4.1. Let G and H be finite graphs.

1. Define the categorical product of G and H to be the graph G×H with vertex set V(G)×V(H)and edge relation given by (v, a) ∼ (w, b) if v ∼G w and a ∼H b.

2. Define the Cartesian product of G and H to be the graph GH with vertex set V(G)×V(H)and edge relation given by (v, a) ∼ (w, b) if one of the following holds

(a) v ∼G w and a = b or(b) v = w and a ∼H b.

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4.4.1 Sabidussi’s theorem

We generalize the Theorem of Sabidussi [Sab57].

Theorem 4.4.2 (Sabidussi). Let G and H be finite graphs. We have the identity

χ(GH) = maxχ(G), χ(H) .

The first step in generalizing this Theorem is to generalize the cartesian product.

Definition 4.4.3. Let J ⊂ Mn and let K ⊂ Mm be submatricial traceless self-adjoint operatorspaces. Let v ⊂ Cn and w ⊂ Cm be bases. Define the cartesian produt of J and K relative to (v, w)as the submatricial traceless self-adjoint operator space

(JK)v,w = J ⊗Dw +Dv ⊗K

where for a basis x = (x1, . . . , xn), Dx = spanxix∗i : i ∈ [n].In the case when e = (e1, . . . , en) and f = (e1, . . . , em), we define the cartesian product JK to

be (JK)e, f .

Lemma 4.4.4. Let G and H be finite graphs with [n] = V(G) and [m] = V(H). We have the identityJGJH = JGH.

Proof. Observe thatJG⊗Dm = spanEv,w⊗Ei,i : v ∼G w, i ∈ [m] and thatDn⊗JH = spanEi,i⊗Ev,w : i ∈ [n], v ∼H w. Combining these, we get that Ei,j ⊗ Ek,l ∈ JGJH if and only if i ∼G j andk = l or i = j and k ∼H l. This is exactly what it means to be a member of JGH.

Lemma 4.4.5. Suppose that J ⊂ Mn and K ⊂ Mm are submatricial traceless self-adjoint operatorspaces. Suppose v ⊂ Cn and w ⊂ Cm are bases. There exist graph homomorphisms J → J ⊗Dw andK → Dv ⊗K. In particular, there exist graph homomorphisms J → (JK)v,w and K → (JK)v,w.

Proof. Define φ : Mn → Mn ⊗Mm : X 7→ 1‖w1‖2 X ⊗ w1w∗1 . This map has Kraus operator E : Cn →

Cn ⊗ Cm : v 7→ v⊗ w1/‖w1‖. Since this Kraus operator is an isometry, we know that φ is cptp.As well, φ(J ) = J ⊗ w1w∗1 ⊂ J ⊗Dw. Similarly, K → Dv ⊗K. Since J ⊗Dw ⊂ (JK)v,w andDv ⊗K ⊂ (JK)v,w, we conclude that J → (JK)v,w and K → (JK)v,w.

Theorem 4.4.6. Let J ⊂ Mn and K ⊂ Mm be submatricial traceless self-adjoint operator spaces. Letv ⊂ Cn and w ⊂ Cm be bases. We have the inequality

maxχ0,cb(J ), χ0,cb(K) ≤ χ0,cb((JK)v,w) .

Proof. By lemma 4.4.5 and by Theorem 4.3.11, we get the inequalities χ0,cb(J ) ≤ χ0,cb((JK)v,w)and χ0,cb(K) ≤ χ0,cb((JK)v,w).

The reverse inequality seems to require the existence of orthogonal bases which colour oursubmatricial traceless self-adjoint operator spaces. The proof mimicks the proof of Sabidussi’sTheorem in [GR16].

Theorem 4.4.7. Let J ⊂ Mn and K ⊂ Mm be submatricial traceless self-adjoint operator spaces. Letc = maxχ0(J ), χ0(K). Suppose that orthonormal bases v ⊂ Cn and w ⊂ Cm exist for which we havemaps f : [n] → [c] and g : [m] → [c] for which whenever f (i) = f (j), v f (i)v∗f (j) ⊥ J and wheneverg(l) = g(k), we have wg(l)w∗g(k) ⊥ K. We have the inequality

χ0((JK)v,w) ≤ maxχ0(J ), χ0(K) .

46

Proof. Let c = maxχ0(J ), χ0(K). Suppose that v, w, f , and g are as above. Define h : [n]× [m]→[c] : (i, j) 7→ f (i) + g(j) mod c. I claim that whenever h(i, j) = h(k, l), that (vi ⊗ wj)(vk ⊗ wl)

∗

is orthogonal to (JK)v,w. The identity h(i, j) = h(k, l) tell us f (i)− f (k) ≡ g(j)− g(l) mod c.If f (i)− f (k) ≡ 0 mod c then we have nothing to check since this means that f (i) = f (k) andg(j) = g(l). Otherwise, viv∗k ⊥ vsv∗s for all s and wjw∗l ⊥ wsw∗s for all s. This guarantees thatviv∗k ⊗ wjw∗l is orthogonal to (JK)v,w.

Remark 7. The same proof as above will show us that for some orthonormal bases v and w,

χ((JK)v,w) ≤ maxχ(J ), χ(K) and

χ((JK)⊥v,w) ≤ maxχ(J ⊥), χ(K⊥) .

We are now ready to state a generalized version of Sabidussi’s theorem. In the followingstatement please recall that χ(X) = χ(X⊥).

Corollary 4.4.8 (Sabidussi’s Theorem for submatricial traceless self-adjoint operator spaces). Sup-pose that J ⊂ Mn and K ⊂ Mm are submatricial traceless self-adjoint operator spaces. There existorthonormal bases v ⊂ Cn and w ⊂ Cm for which we have the inequalities

maxχ0,cb(J ), χ0,cb(K) ≤ χ0,cb((JK)v,w) ≤ χ((JK)⊥v,w) ≤ maxχ(J ⊥), χ(K⊥) .

Proof. By Remark 7, we get the inequality

χ((JK)⊥v,w) ≤ maxχ(J ⊥), χ(K⊥) .

By Theorem 4.3.3 and Corollary 4.3.12, we get the inequality

χ0,cb((JK)v,w) ≤ χst((JK)v,w) ≤ χ((JK)⊥v,w) .

Finally, by Theorem 4.4.6 we get the final inequality.

4.4.2 Hedetniemi’s inequality

The inequality we wish to generalize in this section is a Theorem of Hedetniemi.

Theorem 4.4.9 (Hedetniemi’s inequality). Suppose that G and H are finite graphs. We have the inequality

χ(G× H) ≤ minχ(G), χ(H)

This Theorem follows as a special case of the analogous result for χ0,cb, first we generalize thecategorical product.

Proposition 4.4.10. Let G and H be finite graphs. We have the identity

JG ⊗JH = JG×H .

Proof. Observe that

JG ⊗JH = spanEi,j ⊗ Ek,l : i ∼G j, k ∼H l= JG×H .

47

We now get a generalization of Hedetniemi’s inequality to χ0,cb.

Proposition 4.4.11. Suppose that J ⊂ Mn and K ⊂ Mm are submatricial traceless self-adjoint operatorspaces. We have the inequality

χ0,cb(J ⊗K) ≤ minχ0,cb(J ), χ0,cb(K) .

Proof. The partial trace maps produce graph homomorphisms J ⊗K → K and J ⊗K → J . ByTheorem 4.3.11, we get the inequality.

Remark 8. The long standing conjecture of Hedetneimi asked whether we get the identity

χ(G× H) = minχ(G), χ(H)

for any finite graphs G and H. This was recently resolved in the negative by the remarkable work of YaroslovShitov [Shi19].

48

Chapter 5

Non-local Games

5.1 Introduction

In 1964, Bell showed that local hidden-variable theories, which are classical in nature, cannotexplain all quantum mechanical phenomena [Bel64]. This is obtained by exhibiting a violation of aBell inequality by correlations arising from local measurements on an entangled state. Furthermore,in some instances, it is known that only certain measurements can produce these correlations. Sothrough local measurements not only is it possible to verify that nature is not solely governed byclassical theories, it is also possible to obtain conclusive statistical evidence that a specific quantumstate was present and specific measurements were performed. Results of this nature are oftenreferred to as self-testing (also known as rigidity), first formalized by Mayers and Yao in [MY04].Self-testing has wide reaching applications in areas of theoretical computer science includingcomplexity theory [NV18, FJVY19, NW19], certifiable randomness [VV12], device independentquantum cryptography [ABG+07, VV14], and delegated quantum computation [CGJV19a]. See[SB19] for a comprehensive review. Below we visit five natural questions on the topic of self-testingthat we answer in this paper.

The CHSH game [CHSH69] is the prototypical example of a non-local game. In CHSH, twoseparated players, Alice and Bob, are each provided with a single classical bit, s and t, respectively,chosen uniformly at random by a referee; the players reply with single classical bits a and b to thereferee; and win the game if and only if a⊕ b = s ∧ t. Classically, the players can win the CHSHgame with probability at most 75%. Remarkably, if we allow Alice and Bob to share an entangledstate and employ a quantum strategy, then the optimal winning probability is approximately 85%.For an introduction to non-local games, see [CHTW04].

CHSH is also a canonical example of a self-testing game. Prior to the formalization of self-testingby Mayers and Yao it was already known [SW87, Tsi93] that any optimal quantum strategy forCHSH must be, up to application of local isometries, using the Einstein-Podolsky-Rosen (EPR) state

|ψ〉 = 1√2(|00〉+ |11〉) .

Self-testing can be framed either as an statement about non-local games, Bell inequalities,or more generally correlations. The simple formulation of non-local games make them morestraightforward to implement experimentally and to use in complexity theoretic applications. Forexample, multiprover interactive proofs and their associated complexity classes, e.g., MIP, MIP∗

are built on top of non-local games.CHSH is an instance of a non-pseudo-telepathic game. A pseudo-telepathic game is one that exhibits

quantum advantage (i.e, its quantum value is strictly larger than that of its classical value) and its

49

quantum value is 1. CHSH can also be viewed as a linear constraint system (LCS) game over Z2[CM12]. LCS games are non-local games in which Alice and Bob cooperate to convince the refereethat they have a solution to a system of linear equations. We introduce a new generalization ofCHSH to a family of non-pseudo-telepathic LCS games over Zn for all n ≥ 2. These games resolvethe following questions.

Question 5.1.1. Are there states other than the maximally entangled state that can be self-tested by anon-local game?

To date much has been discovered about self-testing the maximally entangled state, 1√d ∑d−1

j=0 | j〉| j〉.Mermin’s magic square game [Mer90] can be used to self-test two copies of the EPR state and theparallel-repeated magic square game can be used to self-test 2n copies of the EPR state [CN16].

The sum of squares (SOS) decomposition technique in [BP15] shows that the tilted CHSH is aself-test for any pure state of two entangled qubits. This self-testing is stated in terms of violationof Bell inequalities. It is an open problem if the same applies for non-local games. The case for self-testing in higher dimensions has proven more difficult to analyze. Remarkably, it is still possible toself-test any bipartite entangled state, in any dimension [CGS17]. However, these self-test resultsare presented in terms of violations of correlations, unlike the CHSH game which arises from anon-local game (with binary payoff). Our games also resolve in the negative the question “Canevery LCS game be played optimally using the maximally entangled state?” posed in [CM12].

Question 5.1.2. Are there non-local games that provide a self-test for measurements that are not constructedfrom qubit Pauli operators?

The protocols in all of the above examples also provide a self-test for the measurement operators.That is if the players are playing optimally then they must, up to application of local isometries,have performed certain measurements. Self-testing proofs rely on first showing that operators inoptimal strategies must satisfy certain algebraic relations. These relations help identify optimaloperators as representations of some group. This is then used to determine the measurements andstate up to local isometries. In the case of CHSH, one can verify that Alice and Bob’s measurementsmust anti-commute if they are to play optimally. These relations are then enough to conclude thatoperators of optimal strategies generate the dihedral group of degree 4 (i.e., the Pauli group). ThusCHSH is a self-test for the well-known Pauli matrices σX and σZ [MYS12].

Self-tests for measurements in higher dimensions have been primarily focused on self-testingn-fold tensor-products of σX and σZ [NV17, Col16, Mck16]. It is natural to ask if there are self-testsfor operators that are different than ones constructed from qubit Pauli operators. Self-testingClifford observables has also been shown in [CGJV19b]. Our games provides another example thatis neither Pauli nor Clifford. Since our games are LCS this resolves the question, first posed by[CS18], in the affirmative.

Question 5.1.3. Can we extend the solution group formalism for pseudo-telepathic LCS games to a frame-work for proving self-testing for all LCS games?

The solution group introduced in [CLS17] is an indispensable tool for studying pseudo-telepathicLCS games. To each such game there corresponds a group known as the solution group. Optimalstrategies for these games are characterized by their solution group in the sense that any perfectquantum strategy must induce certain representations of this group. Additionally, the work in[CS18] takes this further by demonstrating a streamlined method to prove self-testing certain LCSgames. It is natural to ask whether these methods can be extended to cover all LCS games. Inthis paper we make partial progress in answering this question by introducing a SOS framework,

50

and use it to prove self-testing for our games. At its core, this framework utilizes the interplaybetween sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem[GH17, Vid18] from approximate representation theory.

Question 5.1.4. Is there a systematic approach to design self-tests for arbitrary finite groups?

Informally a game is a self-test for a group if every optimal strategy induces a state dependentrepresentation of the group. In every example that we are aware of, the self-tested solution groupfor pseudo-telepathic LCS games is the Pauli group. Slofstra, in [Slo19], introduced an embeddingtheorem that embeds (almost) any finite group into the solution group of some LCS game. Withthe embedding theorem, the problem of designing games with certain properties reduces tofinding groups with specific properties. Slofstra uses this connection to design games that exhibitseparations between correlation sets resolving the ‘middle’ Tsirelson’s Problem.

However, there are three shortcomings to this approach. Firstly, the resulting game is verycomplex. Secondly, not all properties of the original group are necessarily preserved. Finally, thegame is not a self-test for the original group. Our games self-test an infinite family of groups, nonof which are the Paulis. One such example is the alternating group of degree 4. The SOS frameworkmakes partial progress towards a general theory for self-testing arbitrary groups.

Question 5.1.5. Is there a non-local game that is not a self-test?

In addition to the infinite family of games, we introduce an LCS game that is obtained from“gluing” together two copies of the magic square game. To the best of our knowledge, this gluedmagic square provides the first example of a game that is not a self-test [Mer90].

5.1.1 Main Results

We introduce a family of non-local games Gn defined using the following system of equations overZn

x0x1 = 1,x0x1 = ωn.

We are identifying Zn as a multiplicative group and ωn as the primitive nth root of unity. Note thatthe equations are inconsistent, but this does not prevent the game from being interesting. Aliceand Bob try to convince a referee that they have a solution to this system of equations. Each playerreceives a single bit, specifying an equation for Alice and a variable for Bob, and subsequently eachplayer returns a single number in Zn. Alice’s response should be interpreted as an assignmentto variable x0 in the context of the equation she received, and Bob’s response is interpreted as anassignment to the variable he received. The referee accepts their response iff their assignments areconsistent and satisfy the corresponding equation. The case n = 2 is the CHSH game. The classicalvalue of these games is 3

4 . In Section 5.4, we give a lower-bound on the quantum value of this familyof games. Specifically in Theorem 5.4.9, we show that the quantum value is bounded below by

12+

12n sin

(π2n

) >34

.

We show that the lower-bound is tight in the case of n ≤ 5. We have numerical evidence that theselower-bounds are tight for all n. Specifically, we can find an upper-bound on the quantum value ofa non-local game using the well-known hierarchy of semi-definite programs due to [NPA08]. It is

51

of interest to note that the upper-bound is not obtained using the first level of the NPA hierarchy, asis the case with the CHSH game. Instead, the second level of this hierarchy was needed for n ≥ 3.

The optimal quantum strategy for these games uses the entangled state

|ψn〉 =1

γn

n−1

∑i=0

(1− zn+2i+1)|σi(0), σ−i(0)〉 ∈ HA ⊗HB,

where γn is the normalization factor, σn = (0, 1, . . . , n− 1) is a permutation, and zn is a 4n’th root ofunity. Observe that the state |ψn〉 has full Schmidt rank. Despite this, in all cases except n = 2, thestate |ψn〉 is not the maximally entangled state. For n > 2, the entropy of our state is not maximal,but approaches the maximal entropy of log(n) in the limit.

In Section 5.6, we show that the group generated by the optimal strategy has the followingpresentation

Gn =

⟨P0, P1, J | Pn

0 , Pn1 , Jn, [J, P0], [J, P1], Ji

(Pi

0P−i1

)2for i = 1, 2, . . . , bn/2c

⟩.

For example G3 = Z3 × A4 where A4 is the alternating group of degree 4. We show that our gamesare a self-test for these groups, for n ≤ 5, in the sense that every optimal play of this game inducesa representation of this group. We conjecture that this is true for all n. This partially resolvesQuestion 5.1.4.

In section 5.7, we analyze our game in the case n = 3 and show that it can be used as a robustself-test for the following state

1√10

((1− z4)|00〉+ 2|12〉+ (1 + z2)|21〉

)∈ C3 ⊗C3,

where z := eiπ/6 is the primitive 12th root of unity. Since this state is not the maximally entangledstate, we have thus provided an answer to Question 5.1.1. This game also answers Question 5.1.2since it provides a robust self-test for the following operators

A0 =

0 0 11 0 00 1 0

, A1 =

0 0 −z2

z2 0 00 z2 0

,

B0 =

0 0 11 0 00 1 0

, B1 =

0 −z2 00 0 z2

z2 0 0

,

which do not generate the Pauli group of dimension 3.In Section ??, we introduce the sum of squares framework, using an important lemma proven in

Section 5.2.4, that gives a streamlined method for proving self-testing. We then use this frameworkto prove self-testing for our games. Furthermore, in Section 5.8, we show that when restricted topseudo-telepathic games, the SOS framework reduces to the solution group formalism of Cleve,Liu, and Slofstra [CLS17].

In section 5.9, we construct an LCS game that is obtained from “gluing” two copies of the magicsquare game together. This game is summarized in Figure 5.1.1. We exhibit two inequivalentperfect strategies and thus provide an answer to Question 5.1.5.

52

e1 — e2 — e3| | ||

e4 — e5 — e6| | ||

e7 — e8 — e9||

e10 — e11 — e12|| | |

e13 — e14 — e15|| | |

e16 — e17 — e18

Figure 5.1: This describes an LCS game with 18 variables e1, e2, . . . , e18. Each single-line indicatesthat the variables along the line multiply to 1, and the double-line indicates that the variables alongthe line multiply to −1.

5.1.2 Proof techniques

We prove self-testing in this paper following a recipe that we refer to as the SOS framework. At itscore it applies the Gowers-Hatami (GH) theorem which is a result in approximate-representationtheory. GH has been used previously in proving self-testing, but some of the details have beenoverlooked in the literature. In this paper, we prove Lemma 5.2.4 that encapsulates the use of GH inproving self-testing. In Section 5.2.4, we define approximate representations, irreducible strategies,the Gowers-Hatami theorem and present the proof of the following lemma.

Lemma (informal). Let GA, GB be groups. Suppose every optimal strategy of the game G induces a pairof approximate representations of GA and GB. Further suppose that there is a unique optimal irreduciblestrategy (ρ, σ, |ψ〉) where ρ, σ are irreps of GA, GB, respectively. Then G is a self-test.

Applying this lemma requires us to ascertain two properties of the game:

1. Every optimal strategy induces approximate representations of some groups GA and GB.

2. There is a unique irreducible strategy (ρ, σ, |ψ〉) for the game G.

The first step is to obtain the bias expression for the game G that allows for a simple calculationof the wining probability of any startegy S = (Ai, Bj, |ψ〉) (here Ai and Bj are Alice and Bob’smeasurement observables, respectively, and |ψ〉 is the shared state). The bias expression for Gn isgiven by

Bn(A0, A1, B0, B1) =n−1

∑i=1

Ai0B−i

0 + Ai0Bi

1 + Ai1B−i

0 + ω−i Ai1Bi

1.

Then the winning probability of S is given by ν(G,S) = 〈ψ|( 14nBn(A0, A1, B0, B1) +

1n )|ψ〉. For any

real λ for which there exist some polynomials Tk giving a sum of squares decomposition such as

λI −Bn(A0, A1, B0, B1) = ∑k

T∗k (A0, A1, B0, B1)Tk(A0, A1, B0, B1),

provides an upper bound of λ4n + 1

n on the optimal value of the game (which we denote by ν∗(Gn)).This follows since expressing λI −Bn as an SOS proves that it is a positive semidefinite operatorand consequently 〈ψ|Bn|ψ〉 ≤ λ for all states |ψ〉.

53

Now if we have an SOS for λ = 4nν∗(G) − 4, then we can obtain some algebraic relationsthat every optimal strategy must satisfy. This follows since every optimal strategy must satisfy〈ψ|(λI − Bn)|ψ〉 = 0, from which it follows Tk|ψ〉 = 0 for all k.

Let (Mj(A0, A1)− I)|ψ〉 = 0 be all the relations derived from the SOS relations Tk|ψ〉 = 0 suchthat Mi are monomials only in Alice’s operators, and let GA be the group with the presentation

GA = 〈P0, P1 : Mi(P0, P1)〉

We similarly obtain a group GB for Bob. These are the group referred in the above lemma. For thefirst assumption one must show that any optimal strategy gives approximate representations ofthese groups.

The next step is to prove the second assumption. We need to show that among all the pairsof irreps of GA and GB only one could give rise to an optimal strategy. To this end, we letRi(A0, A1)|ψ〉 = 0 be all the relations derived from relations Tk|ψ〉 = 0. These Ri are allowed to bearbitrary polynomials (as opposed to monomials in the case of group relations). So any optimalirrep must satisfy all these polynomial relations. In some special cases, e.g., games Gn, there is onepolynomial relation that is enough to identify the optimal irreps.

5.1.3 Relation to prior generalizations of CHSH

Much work has been done to generalize CHSH to games over Zn. Initial generalizations weredone by Bavarian and Shor [BS15] and later extended in Kaniewski et al. [KvT+18]. The game wepresent in section 5.3 provides a different generalization by viewing CHSH as an LCS game. Theclassical value of our games is found to be 3

4 from casual observation. Furthermore, we showcasequantum advantage by providing a lower bound on the quantum value for all n.

In contrast the generalization of CHSH discussed in Kaniewski et al. is so difficult to analyzethat even the classical value is not known except in the cases of n = 3, 5, 7. Additionally thequantum value of their Bell inequality is only determined after multiplying by choices of “phase”coefficients. Self-testing for this generalization is examined by Kaniewski et al., where they proveself-testing for n = 3 and show a weaker form of self-testing in the cases of n = 5, 7. For the gameswe introduce, we have self-testing for n = 3, 4, 5 and we conjecture that they are self-tests, in thestrict sense, for all n.

5.1.4 Further work

This paper leaves many open problems and avenues for further investigation. The most importantof these follow.

1. We conjecture that the class of games Gn are rigid for all n. The step missing from resolvingthis conjecture is an SOS decomposition ν(Gn,Sn)I − Bn = ∑k αn,kT∗n,kTn,k for n > 5 wherepolynomials Tn,k viewed as vectors have unit norms and αn,k are positive real numbers.

If this conjecture is true, then we have a simple family of games with 1 bit question and log nbit answer sizes that are self-testing full-Schmidt rank entangled states of any dimension. Infact, we show that the amount of entanglement in these self-tested states rapidly approachesthe maximum amount of entanglement. To the best of our knowledge this is the first exampleof a family of games with such parameters.

2. In Section 5.6, we give efficient explicit presentations for Gn and its multiplication table. Canwe go further and characterize these groups in terms of direct and semidirect products of

54

small well-known groups? The first few cases are as follows

G3 ∼= Z3 × A4, G4∼= (Z3

2 o Z4)o Z4, G5 ∼= (Z42 o Z5)×Z5,

G6 ∼= Z3 ×((((Z4 ×Z3

2)o Z2)o Z2)o Z3)

.

3. The third problem is to characterize all mod n games over two variables and two equations.Let (Zn, m1, m2) be the LCS game mod n based on the system of equations

x0x1 = ωm1n

x0x1 = ωm2n .

So for example (Zn, 0, 1) = Gn. A full characterization includes explicit construction ofoptimal strategies, a proof of self-testing, and a characterization of the group generated byoptimal strategies (i.e., the self-tested group). Interesting observations can be made about thesegames. For example (Z4, 0, 2) self-tests the same strategy as CHSH. Another interestingobservation is that the self-tested group of (Z3, 0, 1) and (Z3, 0, 2) is G3 ∼= Z3 × A4, whereasthe self-tested group of (Z3, 1, 2) is A4.

These games have similar bias expressions to those of Gn. It is likely that the same kind ofmethods can be used to find optimal strategies and establish self-testing for these games.For example (Zn, 0, m) for all m ∈ [n] \ 0 self-test the same group Gn. Just like Gn, therepresentation theory of Gn dictates the optimal strategies of all these games: the optimalirreducible strategies of (Zn, 0, m) for all m ∈ [n] \ 0 are distinct irreps of Gn of degree n.

For example optimal strategies for all games (Z5, 0, m), where m ∈ [5] \ 0, generate G5.This group has 15 irreps of degree five. For each m ∈ [5], there are three irreps sendingJ → ωm

5 I5. For each m ∈ [5] \ 0, the unique optimal irrep strategy of (Z5, 0, m) is one ofthese three irreps.

These games are a rich source of examples for self-testing of groups. A full characterization isa major step toward resolving Question 5.1.4.

4. One drawback of mod n games is that the size of the self-tested groups grows exponentially,|Gn| = 2n−1n2. Where are the games that self-test smaller groups for example the dihedralgroup of degree 5, D5? It seems that to test more groups, we need to widen our search space.

In a similar fashion to mod n games, define games (G, g1, g2) where G is a finite group andg1, g2 ∈ G, based on the system of equations

x0x1 = g1

x0x1 = g2.

Understanding the map that sends (G, g1, g2) to the self-tested group helps us develop aricher landscape of group self-testing.

5. How far can the SOS framework be pushed to prove self-testing? The first step in answeringthis question is perhaps a characterization of games (G, g1, g2) (and their variants, e.g., systemof equations with more variables and equations) using this framework.

6. Glued magic square, as presented in Section 5.9, is not a self-test for any operator solution, butboth inequivalent strategies that we present use the maximally entangled state. Is the gluedmagic square a self-test for the maximally entangled state? If true, this would be the first

55

example of a non-local game that only self-tests the state and not the measurement operators.This positively resolves a question asked in [SB19] in the context of non-local games.

Most self-testing results rely on first attaining self-testing for measurement operators. Soinnovative techniques are needed to prove a state self-testing result for the glued magicsquare game.

5.1.5 Organization of paper

In section 5.2, we fix the nomenclature and give basic definitions for non-local games, winningstrategies, self-testing, LCS games, approximate representation, and the Gowers-Hatami theorem.In section 5.3, we give the generalization of CHSH and derive the bias operator of these games, thatis used in the rest of the paper. In Section 5.4, we establish lower-bounds on the quantum valuefor these games by presenting explicit strategies. In this section we also analyse the entanglemententropy of the shared states in these explicit strategies. In Section 5.6, we give a presentation for thegroups generated by Alice and Bob’s observables. In Section ??, we present the SOS frameworkand give a basic example of its application in proving self-testing. In section 5.7, we use the SOSframework to show that our lower-bound is tight in the case of n = 3, and answer the questions weposed about self-testing. In section 5.8, we show that the SOS framework reduces to the solutiongroup formalism in the case of pseudo-telepathic LCS games. Finally, in Section 5.9 we provide anexample of a non-rigid game.

Acknowledgements

We would like to thank Henry Yuen for having asked many of the motivating questions for thisproject, as well as many insightful discussions and helpful notes on writing this paper. We alsothank the reviewers from QIP for their suggestions and for suggesting the name glued magicsquare.

5.2 Preliminaries

We assume the reader has a working understanding of basic concepts from the field of quantuminformation theory. For an overview of quantum information, refer to [Wat18, CN10, HPP16].

5.2.1 Notation

We use G to refer to a group, while G is reserved for a non-local game. Let [n, m] denote the setn, n+ 1, . . . , m for integers n ≤ m, and the shorthand [n] = [0, n− 1]. This should not be confusedwith [X, Y], which is used to denote the commutator XY−YX. We let In denote the n× n identitymatrix and ei, for i ∈ [n], be the ith standard basis vector. The pauli observables are denoted σx, σy,and σz. The Kronecker delta is denoted by δi,j.

We will letH denote a finite dimensional Hilbert space and use the notation |ψ〉 ∈ H to refer tovectors inH. We use L(H) to denote the set of linear operators in the Hilbert spaceH. We use Un(C)to denote the set of unitary operators acting on the Hilbert space Cn. The set of projection operatorsacting on H are denoted by Proj(H). Given a linear operator A ∈ L(H), we let A∗ ∈ L(H)denote the adjoint operator. For X, Y ∈ L(H), the Hilber-Schmidt inner product is given by〈X, Y〉 = Tr(X∗Y). We also use the following shorthands Trρ(X) = Tr(Xρ) and 〈X, Y〉ρ = Trρ(X∗Y)

56

where X, Y ∈ L(H) and ρ is a density operator acting onH (i.e., positive semidefinite with trace 1).The von Neumann entropy of a density matrix ρ is given by S(ρ) = −Tr(ρ log ρ).

We use <(α) to denote the real part of a complex number α. We let ωn = e2iπ/n be the nth rootof unity. The Dirichlet kernel is Dm(x) = 1

2π ∑mk=−m eikx which by a well known identity is equal to

sin((m+ 12 )x)

2π sin( x2 )

.

The maximally entangled state with local dimension n is given by |Φn〉 = 1√n ∑n−1

i=0 |i〉|i〉 ∈Cn ⊗Cn.

Let HA,HB be Hilbert spaces of dimension n and |ψ〉 ∈ HA ⊗HB be a bipartite state. Thenthere exists orthonormal bases |iA〉n−1

i=0 for HA and |iB〉n−1i=0 for HB and unique non-negative

real numbers λin−1i=0 such that |ψ〉 = ∑n−1

i=0 λi|iA〉|iB〉. The λi’s are known as Schmidt coefficients.The Schmidt rank of a state is the number of non-zero Schmidt coefficients λi. The Schmidt

rank is a rough measure of entanglement. In particular, a pure state |ψ〉 is entangled if and only ifit has Schmidt rank greater than one.

Another measure of entanglement is the entanglement entropy. Given the Schmidt decompositionof a state |ψ〉 = ∑n−1

i=0 λi|iA〉|iB〉, the entanglement entropy Sψ is given by −∑n−1i=0 λ2

i log(λ2i ). The

maximum entanglement entropy is log(n). A pure state is separable (i.e. not entangled) when theentanglement entropy is zero. If the entanglement entropy of a state |ψ〉 is maximum, then the stateis the maximally entangled state up to local unitaries, i.e., there exist unitaries UA, UB ∈ Un(C),such that |ψ〉 = UA ⊗UB|Φn〉.

5.2.2 Non-local games

A non-local game is played between a referee and two cooperating players Alice and Bob who cannotcommunicate once the game starts. The referee provides each player with a question (input), andthe players each respond with an answer (output). The referee determines whether the playerswin with respect to fixed conditions known to all parties. Alice does not know Bob’s question andvice-versa as they are not allowed to communicate once the game starts. However, before the gamestarts, the players could agree upon a strategy that maximizes their success probability. Below wepresent the formal definition and some accompanying concepts.

Definition 5.2.1. A non-local game G is a tuple (IA, IB,OA,OB, π, V) where IA and IB are finitequestion sets, OA and OB are finite answer sets, π denotes the probability distribution on the setIA × IB and V : IA × IB ×OA ×OB → 0, 1 defines the winning conditions of the game.

When the game begins, the referee chooses a pair (i, j) ∈ IA × IB according to the distributionπ. The referee sends i to Alice and j to Bob. Alice then responds with a ∈ OA and Bob with b ∈ OB.The players win if and only if V(i, j, a, b) = 1.

A classical strategy is defined by a pair of functions fA : IA → OA for Alice and fB : IB → OBfor Bob. The winning probability of this strategy is

∑i,j

π(i, j)V(i, j, fA(i), fB(j)).

The classical value, ν(G), of a game is the supremum of this quantity over all classical strategies( fA, fB).

A quantum strategy S for G is given by Hilbert spaces HA, HB, a state |ψ〉 ∈ HA ⊗HB, andprojective measurements Ei,aa∈OA ⊂ Proj(HA) and Fj,bb∈OB ⊂ Proj(HB) for all i ∈ IA andj ∈ IB.

57

Alice and Bob each have access to Hilbert spacesHA andHB respectively. On input (i, j), Aliceand Bob measure their share of the state |ψ〉 according to Ei,aa∈OA and Fj,bb∈OB . The probabilityof obtaining outcome a, b is given by 〈ψ|Ei,a ⊗ Fj,b|ψ〉. The winning probability of strategy S ,denoted by ν(G,S) is therefore

ν(G,S) = ∑i,j,a,b

π(i, j)〈ψ|Ei,a ⊗ Fj,b|ψ〉V(i, j, a, b).

The quantum value of a game, written ν∗(G), is the supremum of the winning probability over allquantum strategies.

The famous CHSH game [CHSH69] is the tuple (IA, IB,OA,OB, π, V) where IA = IB = OA =OB = 0, 1, π is the uniform distribution on IA × IB, and V(i, j, a, b) = 1 if and only if

a + b ≡ ij mod 2.

The CHSH game has a classical value of 0.75 and a quantum value of 12 +

√2

4 ≈ 0.85 [CHSH69].A strategy S is optimal if ν(G,S) = ν∗(G). When a game’s quantum value is larger than the

classical value we say that the game exhibits quantum advantage. A game is pseudo-telepathic if itexhibits quantum advantage and its quantum value is 1.

An order-n generalized observable is a unitary U for which Un = I. It is customary to assign anorder-n generalized observable to a projective measurement system E0, . . . , En−1 as

A =n−1

∑i=0

ωinEi.

Conversely, if A is an order-n generalized observable, then we can recover a projective measurementsystem E0, . . . , En−1 where

Ei =1n

n−1

∑k=0

(ω−i

n A)k

.

In this paper, present strategies in terms of generalized observables.Consider the strategy S consisting of the shared state |ψ〉 ∈ HA ⊗HB and observables Aii∈IA

and Bjj∈IB for Alice and Bob. We say the game G is a self-test for the strategy S if there existε0 ≥ 0 and δ : R+ → R+ a continuous function with δ(0) = 0, such that the following hold

1. S is optimal for G.

2. For any 0 ≤ ε ≤ ε0 and any strategy S = (Aii∈IA , Bjj∈IB , |ψ〉) where |ψ〉 ∈ HA ⊗ HB

and ν(G, S) ≥ ν∗(G)− ε, there exist local isometries VA and VB, and a state |junk〉 such thatthe following hold

•∥∥VA ⊗VB|ψ〉 − |ψ〉|junk〉

∥∥ ≤ δ(ε),

•∥∥VA Ai ⊗VB|ψ〉 − (Ai ⊗ I|ψ〉)|junk〉

∥∥ ≤ δ(ε) for all i ∈ IA,

•∥∥VA ⊗VBBj|ψ〉 − (I ⊗ Bj|ψ〉)|junk〉

∥∥ ≤ δ(ε) for all j ∈ IB.

We use the terminology rigidity and self-testing interchangeably. Exact rigidity is a weaker notion inwhich, we only require the second condition to hold for ε = 0. In Section ??, we give as an examplethe proof of exact rigidity of the CHSH game.

58

5.2.3 Linear constraint system games

A linear constraint system (LCS) game is a non-local game in which Alice and Bob cooperate toconvince the referee that they have a solution to a system of linear equations over Zn. The refereesends Alice an equation and Bob a variable in that equation, uniformly at random. In response,Alice specifies an assignment to the variables in her equation and Bob specifies an assignment tohis variable. The players win exactly when Alice’s assignment satisfies her equation and Bob’sassignment agrees with Alice. It follows that an LCS game has a perfect classical strategy if andonly if the system of equations has a solution over Zn. Similarly the game has a perfect quantumstrategy if and only if the system of equations, when viewed in the multiplicative form, has anoperator solution [CM12].

To each LCS game there corresponds a group referred to as the solution group. The representationtheory of solution group is an indispensable tool in studying pseudo-telepathic LCS games [CLS17,CS18]. In what follows we define these terms formally, but the interested reader is encouraged toconsult the references to appreciate the motivations. In this paper, we are interested in extendingsolution group formalism to general LCS games using the sum of squares approach. We explorethis extension in Section 5.7. When restriced to psuedo-telepathic LCS games, our SOS approach isidentical to the solution group formalism. We present this in section 5.8 for completeness.

Consider a system of linear equations Ax = b where A ∈ Zr×sn , b ∈ Zr

n. We let Vi denote the setof variables occurring in equation i

Vi = j ∈ [s] : ai,j 6= 0.

To view this system of linear equations in multiplicative form, we identify Zn multiplicatively as1, ωn, . . . , ωn−1

n . Then express the ith equation as

∏j∈Vi

xaijj = ωbi

n .

In this paper we only use this multiplicative form. We let Si denote the set of satisfying assignmentsto equation i. In the LCS game GA,b, Alice receives an equation i ∈ [r] and Bob receives a variablej ∈ Vi, uniformly at random. Alice responds with an assignment x to variables in Vi and Bob withan assignment y to his variable j. They win if x ∈ Si and xj = y.

The solution group GA,b associated with GA,b, is the group generated by g1, . . . , gs, J, satisfyingthe relations

1. gnj = Jn = 1 for all j,

2. gj J = Jgj for all j,

3. gjgk = gkgj for j, k ∈ Vi for all i, and

4. ∏j∈Vig

Aijj = Jbi .

5.2.4 Gowers-Hatami theorem and its application to self-testing

In order to precisely state our results about self-testing in Section 5.7, we recall the Gowers-Hatamitheorem and (ε, |ψ〉)-representation [GH17, CS18, Vid18].

59

Definition 5.2.2. Let G be a finite group, n an integer, Hilbert spacesHA,HB of dimension n, and|ψ〉 ∈ HA ⊗HB a state with the reduced density matrix σ ∈ L(HA). An (ε, |ψ〉)-representation ofG, for ε ≥ 0, is a function f : G → Un(C) such that

Ex,y<(〈 f (x)∗ f (y), f (x−1y)〉σ

)≥ 1− ε. (5.2.1)

In the case of ε = 0, we abbreviate and call such a map a |ψ〉-representation, in which case thecondition 5.2.1 simplifies to

〈 f (x)∗ f (y), f (x−1y)〉σ = 1,

or equivalently

f (y)∗ f (x) f (x−1y)|ψ〉 = |ψ〉, (5.2.2)

for all x, y ∈ G. In Condition (5.2.2), we are implicitly dropping the tensor with identity on HB.Note that a |ψ〉-representation f is just a group representation when restricted to the Hilbert spaceH0 = span f (g)|ψ〉 : g ∈ G, i.e., the Hilbert space generated by the image of f acting on |ψ〉. Tosee this, we first rewrite (5.2.2) as

f (x−1y)|ψ〉 = f (x)∗ f (y)|ψ〉.

Thus for any x, y ∈ G we have

f (x−1)∗ f (x−1y)|ψ〉 = f (xx−1y)|ψ〉 = f (y)|ψ〉.

We can multiply both sides by f (x−1) to obtain f (x−1y)|ψ〉 = f (x−1) f (y)|ψ〉 for all x, y ∈ G orequivalently

f (x) f (y)|ψ〉 = f (xy)|ψ〉 for all x, y ∈ G. (5.2.3)

This shows that for all x ∈ G, the operator f (x) leaves the subspace H0 invariant. Thus we can viewf (x)|H0 , the restriction of f (x) to this subspace, as an element of L(H0). Furthermore, by (5.2.3), themap x 7→ f (x)|H0 is a homormorphism and thus a representation of G on H0.

We need the following special case of the Gowers-Hatami (GH) theorem as presented in [Vid18].The analysis of the robust rigidity of these games uses the general statement of GH, using (ε, |ψ〉)-representation. Although skipped in this paper, the tools are in place to analyse the robust case.

Theorem 5.2.3 (Gowers-Hatami). Let d be an integer, |ψ〉 ∈ Cd ⊗Cd a bipartite state, G a finite group,and f : G → Ud(C) a |ψ〉-representation. Then there exist d′ ≥ d, a representation g : G → Ud′(C), andan isometry V : Cd → Cd′ such that f (x)⊗ I|ψ〉 = V∗g(x)V ⊗ I|ψ〉.

From the proof of this theorem in [Vid18], we can take g = ⊕ρ Id ⊗ Idρ⊗ ρ where ρ ranges over

irreducible representations of G and dρ is the dimension of ρ. Additionally, in the same bases,we can factorize V into a direct sum over irreps such that Vu = ⊕ρ(Vρu), for all u ∈ Cd whereVρ ∈ L(Cd, Cd ⊗Cdρ ⊗Cdρ) are some linear operators. It holds that ∑ρ V∗ρ Vρ = V∗V = Id.

In some special cases, such as in our paper, we can restrict g to be a single irreducible represen-tation of G. In such cases we have a streamlined proof of self-testing. Lemma 5.2.4 below captureshow GH is applied in proving self-testing in these cases.

Let G = (IA, IB,OA,OB, π, V) be a game, GA and GB be groups with generators Pii∈IA andQjj∈IB , GA and GB be free groups over Pii∈IA and Qjj∈IB , and S = (Ai, Bj, |ψ〉) be a

60

strategy where |ψ〉 ∈ CdA ⊗CdB . We define two functions f SA : GA → UdA(C), f SB : GB → UdB(C)

where f SA (Pi) = Ai and f SB (Qj) = Bj and they are extended homomorphically to all of GA andGB, respectively. Suppose that the game G has the property that for every optimal strategy S =

(Ai, Bj, |ψ〉), f SA and f SB are |ψ〉-representations for GA and GB, respectively.Now applying GH, for every optimal strategy S , there exist representations gA, gB of GA, GB,

respectively, and isometries VA, VB such that

f SA (x)⊗ I|ψ〉 = V∗AgA(x)VA ⊗ I|ψ〉 for all x ∈ GA,

I ⊗ f SB (y)|ψ〉 = I ⊗V∗B gB(y)VB|ψ〉 for all y ∈ GB.

Unfortunately this is not enough to establish rigidity for G as defined in Section 5.2.2. To do this,we need and extra assumption on G that we deal with in the following lemma.

For any pair of representations ρ, σ of GA, GB respectively, and state |ψ〉 ∈ Cdσ ⊗ Cdρ , letSρ,σ,|ψ〉 = (ρ(Pi)i∈IA , σ(Qj)j∈IB , |ψ〉) be the strategy induced by the pair of representations(ρ, σ). Also let ν(G, ρ, σ) = max|ψ〉 ν(G,Sρ,σ,|ψ〉).

Lemma 5.2.4. Suppose that there is only one pair of irreps ρ, σ for which ν(G, ρ, σ) = ν∗(G). Additionallyassume that |ψ〉 is the unique state (up to global phase) for which Sρ,σ,|ψ〉 is an optimal strategy. Let

S = (Ai, Bj, |ψ〉) be an optimal strategy of G such that |ψ〉 ∈ CdA ⊗ CdB , f SA and f SB are |ψ〉-representations for GA and GB, respectively. Then there exist isometries VA : CdA → CdA|GA|, VB : CdB →CdB|GB|, and a state |junk〉 such that

VA ⊗VB|ψ〉 = |junk〉|ψ〉,VA Ai ⊗VB|ψ〉 = |junk〉ρ(Pi)⊗ Idσ

|ψ〉,VA ⊗VBBj|ψ〉 = |junk〉Idρ

⊗ σ(Qj)|ψ〉,

for all i ∈ IA, j ∈ IB.

Proof. For simplicity, we only prove the case of binary games, i.e., we assume |OA| = |OB| = 2.The general case follows similarly. For binary games we only need to consider strategies comprisedof binary observables (A is a binary observable if it is Hermitian and A2 = I). Without loss ofgenerality, we can assume that there exist some complex numbers λij, λi, λj, λ such that for anystrategy S = (Ai, Bj, |ψ〉)

ν(G,S) = 〈ψ|(

∑i∈IA,j∈IB

λij Ai ⊗ Bj + ∑i∈IA

λi Ai ⊗ I + ∑j∈IB

λj I ⊗ Bj + λI ⊗ I

)|ψ〉. (5.2.4)

As argued earlier, by GH, we have

f SA (x)⊗ I|ψ〉 = V∗AgA(x)VA ⊗ I|ψ〉, (5.2.5)

I ⊗ f SB (x)|ψ〉 = I ⊗V∗B gB(x)VB|ψ〉, (5.2.6)

where gA = ⊕ρ IdAdρ⊗ ρ, gB = ⊕σ IdBdσ

⊗ σ, where ρ and σ range over irreducible representations ofGA and GB, respectively. We also have the factorization VAu = ⊕ρ(VA,ρu), for all u ∈ CdA as wellas VBu = ⊕σ(VB,σu), for all u ∈ CdB . As mentioned above in the discussion that followed Theorem5.2.3, VA,ρ and VB,σ are some linear operators for which ∑ρ V∗A,ρVA,ρ = IdA and ∑σ V∗B,σVB,σ = IdB .

61

We want to write the winning probability of S in terms of the winning probabilities of irrepstrategies. To this end, let

pρ,σ = ‖VA,ρ ⊗VB,σ|ψ〉‖2,

|ψρ,σ〉 =

1√pρ,σVA,ρ ⊗VB,σ|ψ〉 pρ,σ > 0,

0 pρ,σ = 0,

and consider strategies

SI⊗ρ,I⊗σ,|ψρ,σ〉 = (IdAdρ⊗ ρ(Pi), IdBdσ

⊗ σ(Qj), |ψρ,σ〉).

Using (5.2.4), we can write

ν(G, S) = 〈ψ|(

∑i∈IA,j∈IB

λij Ai ⊗ Bj + ∑i∈IA

λi Ai ⊗ I + ∑j∈IB

λj I ⊗ Bj + λI ⊗ I

)|ψ〉

= ∑ρ,σ〈ψ|V∗A,ρ ⊗V∗B,σ

(∑

i∈IA,j∈IB

λij(IdAdρ⊗ ρ(Pi))⊗ (IdBdσ

⊗ σ(Qj)) + ∑i∈IA

λi(IdAdρ⊗ ρ(Pi))⊗ I

+ ∑j∈IB

λj I ⊗ (IdBdσ⊗ σ(Qj)) + λI ⊗ I

)VA,ρ ⊗VB,σ|ψ〉

= ∑ρ,σ

pρ,σν(G,SI⊗ρ,I⊗σ,|ψρ,σ〉).

Note that ∑ρ,σ pρ,σ = 1. In other words, the winning probability of S is a convex combina-tion of the winning probabilities of irreducible strategies SI⊗ρ,I⊗σ,|ψρ,σ〉. It is easily verified thatν(G,SI⊗ρ,I⊗σ,|ψρ,σ〉) ≤ ν(G, ρ, σ). By assumption of the lemma ν(G, ρ, σ) < ν∗(G) except when

(ρ, σ) = (ρ, σ). Now since S is an optimal strategy, we have

pρ,σ =

1 (ρ, σ) = (ρ, σ),0 otherwise.

Therefore ν(G, S) = ν(G,SI⊗ρ,I⊗σ,|ψρ,σ〉) and hence SI⊗ρ,I⊗σ,|ψρ,σ〉 is an optimal strategy. Fromthe assumption of the lemma ,|ψ〉 is the unique state optimizing the strategy induced by (ρ, σ).Therefore |ψρ,σ〉 = |junk′〉|ψ〉 where both |junk′〉 and |ψ〉 are shared between Alice and Bob suchthat |junk′〉 is the state of the register upon which the identities of Alice and Bob in the operators(I ⊗ ρ)A ⊗ (I ⊗ σ)B are applied. In summary

|ψρ,σ〉 =|junk′〉|ψ〉 (ρ, σ) = (ρ, σ),0 otherwise.

(5.2.7)

Now using (5.2.5), it follows that

Ai ⊗VB|ψ〉 = V∗AgA(Pi)VA ⊗VB|ψ〉,

from which

VA Ai ⊗VB|ψ〉 = VAV∗AgA(Pi)VA ⊗VB|ψ〉.

62

Since VAV∗A is a projection and VA Ai ⊗VB|ψ〉 and gA(Pi)VA ⊗VB|ψ〉 are both unit vectors, it holdsthat

VA Ai ⊗VB|ψ〉 = gA(Pi)VA ⊗VB|ψ〉=⊕ρ,σ

(IdAdρ⊗ ρ(Pi))⊗ IdBd2

σ|ψρ,σ〉

=(|junk′〉ρ(Pi)⊗ Idσ

|ψ〉)⊕(ρ,σ) 6=(ρ,σ) 0dAd2

ρdBd2σ

= |junk〉ρ(Pi)⊗ Idσ|ψ〉,

where the third equality follows from (5.2.7), and in the fourth equality |junk〉 = |junk′〉 ⊕ 0 where

0 ∈ CdAdB(

|GA ||GB |dρdσ

−dρdσ). Note that dAdB(|GA||GB|

dρdσ− dρdσ) is a positive integer because the degree of

an irreducible representation divides the order of the group.

Corollary 5.2.5. If in addition to the assumptions of Lemma 5.2.4, it holds that for every optimal strategyS = (Ai, Bj, |ψ〉), f SA and f SB are |ψ〉-representations, then G is a self-test for the strategy Sρ,σ,|ψ〉.

Note that all these results can be stated robustly using the notion of (ε, |ψ〉)-representation, butin this paper we focus our attention on exact rigidity. In this paper we use SOS to obtain the extraassumption of Corollary 5.2.5 as seen in Sections ?? and 5.7.

5.3 A generalization of CHSH

The CHSH game can also be viewed as an LCS game where the linear system, over multiplicativeZ2, is given by

x0x1 = 1,x0x1 = −1.

The CHSH viewed as an LCS is first considered in [CM12]. We generalize this to a game Gn overZn for each n ≥ 2

x0x1 = 1,x0x1 = ωn.

As is the case for G2 = CHSH, the classical value of Gn is easily seen to be 0.75. In Section 5.4,we exhibit quantum advantage by presenting a strategy Sn showing that ν∗(Gn) ≥ ν(Gn,Sn) =12 +

12n sin( π

2n )> 1

2 +1π ≈ 0.81. In Section 5.6, we present the group Gn generated by the operators

in Sn. In Section 5.7, we show that G3 is a self-test, and conjecture that this is true for all n ≥ 2.As defined in the preliminaries, conventionally, in an LCS game, Alice has to respond with an

assignment to all variables in her equation. It is in Alice’s best interest to always respond with asatisfying assignment. Therefore, the referee could always determine Alice’s assignment to x1 fromher assignment to x0. Hence, without loss of generality, in our games, Alice only responds with anassignment to x0.

63

Formally Gn = ([2], [2], Zn, Zn, π, V) where Zn = 1, ωn, . . . , ωn−1n , π is the uniform distribu-

tion on [2]× [2], and

V(0, 0, a, b) = 1 ⇐⇒ a = b,V(0, 1, a, b) = 1 ⇐⇒ ab = 1,V(1, 0, a, b) = 1 ⇐⇒ a = b,V(1, 1, a, b) = 1 ⇐⇒ ab = ωn.

Consider the quantum strategy S given by the state |ψ〉, and projective measurements E0,aa∈[n]and E1,aa∈[n] for Alice, and F0,bb∈[n] and F1,bb∈[n] for Bob. Note that in our measurementsystems, we identify outcome a ∈ [n] with answer ωa

n ∈ Zn. As done in the preliminaries, define thegeneralized observables A0 = ∑n−1

i=0 ωinE0,i, A1 = ∑n−1

i=0 ωinE1,i, B0 = ∑n−1

i=0 ωinF0,i, B1 = ∑n−1

i=0 ωinF1,i.

We derive an expression for the winning probability of this strategy in terms of the these generalizedobservables. We do so by introducing the bias operator

Bn = Bn(A0, A1, B0, B1) =n−1

∑i=1

Ai0B−i

0 + Ai0Bi

1 + Ai1B−i

0 + ω−in Ai

1Bi1,

in which we dropped the tensor product symbol between Alice and Bob’s operators.

Proposition 5.3.1. Given the strategy S above, it holds that ν (Gn,S) = 14n 〈ψ|Bn|ψ〉+ 1

n .

Proof.

Bn + 4I =n−1

∑i=0

Ai0B−i

0 + Ai0Bi

1 + Ai1B−i

0 + ω−in Ai

1Bi1

=n−1

∑i=0

n−1

∑a,b=0

ωi(a−b)n E0,aF0,b + ω

i(a+b)n E0,aF1,b + ω

i(a−b)n E1,aF0,b + ω

i(a+b−1)n E1,aF1,b

=n−1

∑a,b=0

n−1

∑i=0

ωi(a−b)n E0,aF0,b + ω

i(a+b)n E0,aF1,b + ω

i(a−b)n E1,aF0,b + ω

i(a+b−1)n E1,aF1,b

= nn−1

∑a=0

E0,aF0,a + E0,aF1,−a + E1,aF0,a + E1,aF1,1−a

in which in the last equality we used the identity 1 + ωn + . . . + ωn−1n = 0. Also note that in F1,−a

and F1,1−a second indices should be read mod n. Finally notice that

ν(G,S) = 14〈ψ|

(n−1

∑a=0

E0,aF0,a + E0,aF1,−a + E1,aF0,a + E1,aF1,1−a

)|ψ〉.

5.4 Strategies for Gn

In this section, we present quantum strategies Sn for Gn games. In Section 5.4.2, we show thatν(Gn,Sn) =

12 +

12n sin( π

2n )and that this value approaches 1

2 +1π from above as n tends to infinity.

This lower bounds the quantum value ν∗(Gn), and proves that these games exhibit quantum

64

advantage with a constant gap > 1π −

14 . We also show that the states in these strategies have

full-Schmidt rank. Furthermore the states tend to the maximally entangled state as n→ ∞.We conjecture that Sn are optimal and that the games Gn are self-tests for Sn. In Section 5.7, we

prove this for n = 3. Using the NPA hierarchy, we verify the optimality numerically up to n = 7. Ifthe self-testing conjecture is true, we have a family of games with one bit questions and log(n) bitsanswers, that self-test entangled states of local dimension n for any n.

5.4.1 Definition of the strategy

Let σn = (0 1 2 . . . n − 1) ∈ Sn denote the cycle permutation that sends i to i + 1 mod n. Letzn = ω1/4

n = eiπ/2n. Let Dn,j = In − 2eje∗j be the diagonal matrix with −1 in the (j, j) entry, and1 everywhere else in the diagonal. Then let Dn,S := ∏j∈S Dn,j, where S ⊂ [n]. Finally, let Xn bethe shift operator (also known as the generalized Pauli X), i.e., Xnei = eσn(i). For convenience,we shall often drop the n subscript when the dimension is clear from context, and so just refer tozn, Dn,j, Dn,S, Xn as z, Dj, DS, X, respectively.

LetHA = HB = Cn. Then Alice and Bob’s shared state in Sn is defined to be

|ψn〉 =1

γn

n−1

∑i=0

(1− zn+2i+1)|σi(0), σ−i(0)〉 ∈ HA ⊗HB,

where γn =√

2n + 2sin( π

2n )is the normalization factor. The generalized observables in Sn are

A0 = X

A1 = z2D0XB0 = X

B1 = z2D0X∗.

Example 5.4.1. In S2, Alice and Bob’s observables are

A0 = σx =

(0 11 0

), A1 = σy =

(0 −ii 0

),

B0 = σx =

(0 11 0

), B1 = σy =

(0 −ii 0

),

and their entangled state is

|ψ2〉 =1√

4 + 2√

2

((1 +

1− i√2

)|00〉 −

(1 +

1 + i√2

)|11〉

).

One can verify that this indeed give us the quantum value for CHSH 12 +

√2

4 .

Example 5.4.2. In S3, Alice and Bob’s observables are

A0 =

0 0 11 0 00 1 0

, A1 =

0 0 −z2

z2 0 00 z2 0

,

B0 =

0 0 11 0 00 1 0

, B1 =

0 −z2 00 0 z2

z2 0 0

,

65

with the entangled state

|ψ3〉 =1√10

((1− z4)|00〉+ 2|12〉+ (1 + z2)|21〉

).

One can compute that 〈ψ|B3|ψ〉 = 6. Hence, by Proposition 5.3.1, we have ν∗(G3) ≥ 56 .

5.4.2 Analysis of the strategy

In this section, we prove that Sn is a quantum strategy and calculate its winning probability. Wethen prove that the entanglement entropy of |ψn〉 approaches the maximum entropy as n tends toinfinity.

Proposition 5.4.3. For n ∈N, it holds that ∑n−1j=0 z2j+n+1

n = ∑n−1j=0 z−(2j+n+1)

n .

Proof. A direct computation gives

n−1

∑j=0

z2j+n+1 =2zn+1

1− z2 =2z−n−1

1− z−2 =n−1

∑j=0

z−(2j+n+1),

where we have used the fact that z2n = −1.

Proposition 5.4.4. For n ∈N, it holds that ∑n−1j=0 z2j+n+1

n = − 1sin( π

2n ).

Proof. We handle the even and odd case separately, and in both cases we use the well-knownidentity for the Dirichlet kernel mentioned in preliminaries. For odd n

−n−1

∑j=0

z2j+n+1 =n−1

∑j=0

z2j−(n−1) =

n−12

∑j=− n−1

2

z2j =

n−12

∑j=− n−1

2

eπijn

= 2πD n−12

(π

n

)=

sin(( n−1

2 + 12

)πn

)sin(

π2n

) =1

sin(

π2n

) .

For even n

−n−1

∑j=0

z2j+n+1 = zn

∑j=0

z2j−n − zn+1 = zn2

∑j=− n

2

z2j − zn+1 = 2πzD n2

(π

n

)− zn+1

=(

cos( π

2n

)+ i sin

( π

2n

)) sin(( n

2 + 12

)πn

)sin(

π2n

) − i(

cos( π

2n

)+ i sin

( π

2n

))=

cos2 ( π2n

)+ sin2 ( π

2n

)sin(

π2n

) =1

sin(

π2n

) .

Now let’s observe a commutation relation between Dj and Xk.

Proposition 5.4.5. XiDj = Dσi(j)Xi, for all i, j ∈ [n].

66

Proof. It suffices to prove XDj = Dσ(j)X. We show this by verifying XDjek = Dσ(j)Xek for allk ∈ [n].

XDjek = (−1)δj,k eσ(k) = (−1)δσ(j),σ(k)eσ(k) = Dσ(j)Xek

Now we prove the strategy defined in section 5.4.1 is a valid quantum strategy.

Proposition 5.4.6. A0, A1, B0, B1 are order-n generalized observables and |ψn〉 is a unit vector.

Proof. Observe thatAn

0 = Bn0 = Xn = I,

alsoAn

1 = (z2D0X)n = z2nD0,σ1(0),...,σn−1(0)Xn = (−1)(−I)I = I.

Similarly,Bn

1 = (z2D0X∗)n = z2n(X∗)nD0,σ1(0),...,σn−1(0) = (−1)I(−I) = I.

It is an easy observation that these operators are also unitary. To see that |ψn〉 is a unit vectorwrite

n−1

∑i=0|1− zn+2i+1|2 =

n−1

∑i=0

(1− cos

(π(n + 2i + 1)

2n

))2

+ sin(

π(n + 2i + 1)2n

)2

=n−1

∑i=0

2(

1− cos(

π(n + 2i + 1)2n

))= 2n−

n−1

∑i=0<(zn+2i+1)

= 2n +2

sin(π/2n)= γ2

n,

where we have used Proposition 5.4.4 in the third equality.

Lemma 5.4.7. The entangled state |ψ〉 is an eigenvector for the bias B = ∑n−1j=1 Aj

0B−j0 + Aj

0Bj1 + Aj

1B−j0 +

z−4j Aj1Bj

1 with eigenvalue 2n− 4 + 2sin( π

2n ).

Proof. For the sake of brevity, we drop the normalization factor γn in the derivation below, and let|ϕ〉 = γn|ψn〉. We write

B|ϕ〉 =(

n−1

∑j=1

Aj0 ⊗ B−j

0 + Aj0 ⊗ Bj

1 + Aj1 ⊗ B−j

0 + z−4j Aj1 ⊗ Bj

1

)|ϕ〉

=

(n−1

∑j=1

(X⊗ X∗)j + z2j (X⊗ D0X∗)j + z2j(D0X⊗ X∗)j + (D0X⊗ D0X∗)j

)|ϕ〉.

67

Lemma 5.4.8. (X⊗ D0X∗)j |ϕ〉 = (D0X⊗ X∗)j|ϕ〉 and (X⊗ X∗)j |ϕ〉 = (D0X⊗ D0X∗)j|ϕ〉.

Proof. It suffices to show these identities for j = 1 on states |σi(0), σ−i(0)〉, for all i, in place of |ϕ〉.The result then follows by simple induction. In other words, we prove

(X⊗ D0X∗) |σi(0), σ−i(0)〉 = (D0X⊗ X∗)|σi(0), σ−i(0)〉,(X⊗ X∗) |σi(0), σ−i(0)〉 = (D0X⊗ D0X∗)|σi(0), σ−i(0)〉.

Note that I ⊗ D0|σi+1(0), σ−i−1(0)〉 = D0 ⊗ I|σi+1(0), σ−i−1(0)〉 since −i− 1 = 0 mod n iff i + 1 =0 mod n. Therefore

(X⊗ D0X∗) |σi(0), σ−i(0)〉 = (I ⊗ D0) |σi+1(0), σ−i−1(0)〉= (D0 ⊗ I) |σi+1(0), σ−i−1(0)〉= (D0X⊗ X∗)|σi(0), σ−i(0)〉.

The other identity follows similarly.

Now we write

B|ϕ〉 = 2

(n−1

∑j=1

(X⊗ X∗)j + z2j(D0X⊗ X∗)j

)|ϕ〉

= 2n−1

∑j=1

(1 + z2j(D[j] ⊗ I)

)(X⊗ X∗)j|ϕ〉

= 2n−1

∑j=1

n−1

∑i=0

(1− z2i+n+1

) (1 + z2j(D[j] ⊗ I)

)(X⊗ X∗)j|σi(0), σ−i(0)〉

= 2n−1

∑j=1

n−1

∑i=0

(1− z2i+n+1

) (1 + z2j(D[j] ⊗ I)

)|σi+j(0), σ−(i+j)(0)〉,

where in the second equality we use Proposition 5.4.5, and in the third equality we just expanded|ϕ〉. Note that

(D[j] ⊗ I)|σi+j(0), σ−(i+j)(0)〉 =−|σi+j(0), σ−(i+j)(0)〉 i ∈ [n− j, n− 1],|σi+j(0), σ−(i+j)(0)〉 i ∈ [0, n− j− 1],

and we use this to split the sum

B|ϕ〉 = 2n−1

∑j=1

(n−j−1

∑i=0

(1− z2i+n+1

) (1 + z2j

)|σi+j(0), σ−(i+j)(0)〉

+n−1

∑i=n−j

(1− z2i+n+1

) (1− z2j

)|σi+j(0), σ−(i+j)(0)〉

)

= 2n−1

∑i=0

(n−i−1

∑j=1

(1− z2i+n+1

) (1 + z2j

)|σi+j(0), σ−(i+j)(0)〉

+n−1

∑j=n−i

(1− z2i+n+1

) (1− z2j

)|σi+j(0), σ−(i+j)(0)〉

),

68

and make a change of variable r = i + j to get

B|ϕ〉 = 2n−1

∑i=0

(n−1

∑r=i+1

(1− z2i+n+1

) (1 + z2(r−i)

)|σr(0), σ−r(0)〉

+n+i−1

∑r=n

(1− z2i+n+1

) (1− z2(r−i)

)|σr(0), σ−r(0)〉

).

We have z2(r−i) = z2(r−n+n−i) = z2nz2(r−n−i) = −z2(r−n−i) and σr(0) = σr+n(0), so by anotherchange of variable in the second sum where we are summing over r = [n, n + i− 1] we obtain

B|ϕ〉 = 2n−1

∑i=0

(n−1

∑r=i+1

(1− z2i+n+1

) (1 + z2(r−i)

)|σr(0), σ−r(0)〉

+i−1

∑r=0

(1− z2i+n+1

) (1 + z2(r−i)

)|σr(0), σ−r(0)〉

)

= 2n−1

∑i=0

(n−1

∑r=0

(1− z2i+n+1

) (1 + z2(r−i)

)|σr(0)σ−r(0)〉 − 2

(1− z2i+n+1

)|σi(0)σ−i(0)〉

)

= 2n−1

∑i=0

(n−1

∑r=0

(1− z2i+n+1

) (1 + z2(r−i)

)|σr(0)σ−r(0)〉

)− 4|ϕ〉

= 2n−1

∑r=0|σr(0)σ−r(0)〉

(n−1

∑i=0

(1− z2i+n+1

) (1 + z2(r−i)

))− 4|ϕ〉.

We also have

n−1

∑i=0

(1− z2i+n+1

) (1 + z2(r−i)

)=

n−1

∑i=0

1− z2r+n+1 + z2(r−i) − z2i+n+1

=n−1

∑i=0

1− z2r+n+1 + z2(r−i) − z−(2i+n+1)

= (1− z2r+n+1)n−1

∑i=0

1− z−(2i+n+1)

=

(n +

1sin( π

2n )

)(1− z2r+n+1),

where in the second and last equality we used Propositions 5.4.3 and 5.4.4, respectively. Puttingthese together, we obtain

B|ϕ〉 = 2(

n +1

sin( π2n )

) n−1

∑r=0

(1− z2r+n+1)|σr(0)σ−r(0)〉 − 4|ϕ〉

=

(2n− 4 +

2sin( π

2n )

)|ϕ〉.

69

2 10 20 30 400.81

0.82

0.83

0.84

0.85

0.86

n

ν(G

n,S

n)

2 10 20 30 400.99

0.992

0.994

0.996

0.998

1

n

S ψn/

log(

n)

Figure 5.2: The figure on the left illustrates the fast convergence rate of the winning probabilities asthey approach the limit 1/2 + 1/π. The figure on the right illustrates the ratio of the entanglemententropy to the maximum entanglement entropy of the states for n ≤ 40.

Next we calculate ν(Gn,Sn), its limit as n grows and the entanglement entropy of states |ψn〉.See Figure 5.2.

Theorem 5.4.9. ν(Gn,Sn) =12 +

12n sin( π

2n ).

Proof.

ν(Gn,Sn) =1

4n〈ψ|B|ψ〉+ 1

n

=1

4n〈ψ|

(2n− 4 +

2sin(

π2n

)) |ψ〉+ 1n

=1

4n

(2n− 4 +

2sin(

π2n

))+1n

=12+

12n sin

(π2n

) .

Theorem 5.4.10. The following hold

1. limn→∞ ν(Gn,Sn) = 1/2 + 1/π.

2. ν(Gn,Sn) is a strictly decreasing function.

3. The games Gn exhibit quantum advantage, i.e., for n > 1

ν∗(Gn) > 1/2 + 1/π > 3/4 = ν(Gn).

Proof. For the first statement, it suffices to see that

limx→∞

12x sin

(π2x

) = limx→∞

12x

sin(

π2x

) = limx→∞

−12x2

−π cos( π2x )

2x2

=1π

.

70

For the second statement, we show that the function f (x) = 2x sin(π/2x) is strictly increasingfor x ≥ 1. We have f ′(x) = 2 sin(π/2x) − π cos(π/2x)/x. Then f ′(x) > 0 is equivalent totan(π/2x) ≥ π/2x. This latter statement is true for all x ≥ 1. The third statement follows from thefirst two.

Theorem 5.4.11. States |ψn〉 have full Schmidt rank and the ratio of entanglement entropy to maximumentangled entropy, i.e., Sψn / log(n) approaches 1 as n → ∞. In particular, up to local isometries, thesestates approach the maximally entangled state.

Proof. Recall that

|ψn〉 =1

γn

n−1

∑i=0

(1− z2i+n+1

)|σi(0), σ−i(0)〉 ∈ HA ⊗HB.

Let |iA〉 = 1−z2i+n+1

‖1−z2i+n+1‖ |σi(0)〉 and |iB〉 = |σ−i(0)〉. Clearly iAi and iBi are orthonormal bases for

HA andHB, respectively. The Schmidt decomposition is now given by

|ψn〉 =1

γn

n−1

∑i=0

∥∥∥1− z2i+n+1∥∥∥ |iAiB〉.

To calculate the limit of Sψn / log(n) first note that

Sψn

log(n)= −

∑n−1i=0

∥∥1− z2i+n+1∥∥2 log ‖1−z2i+n+1‖2

γ2n

γ2n log(n)

= −∑n−1

i=0

∥∥1− z2i+n+1∥∥2(

log∥∥1− z2i+n+1

∥∥2 − log γ2n

)γ2

n log(n)

≥ −log(4)∑n−1

i=0

∥∥1− z2i+n+1∥∥2

γ2n log(n)

+log γ2

n ∑n−1i=0

∥∥1− z2i+n+1∥∥2

γ2n log(n)

= − log(4)log(n)

+log γ2

nlog(n)

where for the inequality we used the fact that∥∥1− z2i+n+1

∥∥ ≤ 2, and for the last equality we used

the identity γ2n = ∑n−1

i=0

∥∥1− z2i+n+1∥∥2. So it holds that

− log(4)log(n)

+log γ2

nlog(n)

≤Sψn

log(n)≤ 1.

By simple calculus limn→∞log γ2

nlog(n) −

log(4)log(n) = 1. Therefore by squeeze theorem limn→∞

Sψnlog(n) = 1.

5.5 Group structure of Sn

Let Hn = 〈A0, A1〉 be the group generated by Alice’s observables in Sn. Note that since (A1A∗0)2 =

z4n I, we could equivalently define Hn = 〈A0, A1, z4

n I〉. Also let

Gn =

⟨P0, P1, J | Pn

0 , Pn1 , Jn, [J, P0], [J, P1], Ji

(Pi

0P−i1

)2for i = 1, 2, . . . , bn/2c

⟩.

71

In this section we show that Hn ∼= Gn. So it also holds that Hn is a representation of Gn.We conjecture that Gn is a self-test for Gn, in the sense that every optimal strategy of Gn is a|ψ〉-representation of Gn. In Section 5.7, we prove this for n = 3.

Remark 5.5.1. Note that the relations Ji(

Pi0P−i

1

)2holds in Gn for all i.

The following lemma helps us develop a normal form for elements of Gn.

Lemma 5.5.2. For all i, j, the elements Pi0P−i

1 and Pj0P−j

1 commute.

Proof. (Pi

0P−i1

)(Pj

0P−j1

)= J−iPi

1P−i0 Pj

0P−j1

= J−iPi1Pj−i

0 P−j1

= J−iPi1(

Pj−i0 P−(j−i)

1

)P−i

1

= J−i−(j−i)Pi1Pj−i

1 P−(j−i)0 P−i

1

= J−j(Pj1P−j

0

)(Pi

0P−i1

)= J−j(J jPj

0P−j1

)(Pi

0P−i1

)=(

Pj0P−j

1

)(Pi

0P−i1

).

Lemma 5.5.3. For every g ∈ Gn there exist i, j ∈ [n] and qk ∈ 0, 1 for k = 1, 2, . . . , n− 1 such that

g = JiPj0

(P0P−1

1

)q1(

P20 P−2

1

)q2 · · ·(

Pn−10 P−(n−1)

1

)qn−1 .

Proof. First note that J is central, therefore we can write g in Gn as

g = JiPj10 Pj2

1 Pj30 · · · P

jk1 ,

for some k ∈N, i ∈ [n], jl ∈ [n] where l = 1, 2, . . . , k. Without loss of generality, let k be even. Weperform the following sequence of manipulations

g = JiPj10 Pj2

1 Pj30 · · · P

jk−21 Pjk−1

0 Pjk1

= JiPj10 Pj2

1 Pj30 · · · P

jk−21 Pjk−1

0 Pjk0

(P−jk

0 Pjk1

)= JiPj1

0 Pj21 Pj3

0 · · · Pjk−21 Pjk−1+jk

1

(P−(jk−1+jk)

1 Pjk−1+jk0

)(P−jk

0 Pjk1

)= Ji−(jk−1+jk)Pj1

0 Pj21 Pj3

0 · · · Pjk−2+jk−1+jk1

(P−(jk−1+jk)

0 Pjk−1+jk1

)(P−jk

0 Pjk1

)= · · ·= Ji−sP−s1

0

(Ps2

0 P−s21

)· · ·(

Psk−10 P−sk−1

1

)(Psk

0 P−sk1

),

where sl = −∑kt=l jt and s = −∑(k−2)/2

t=1 s2t+1. Then we use the commutation relationship fromlemma 5.6.2 to group the terms with the same P0 and P1 exponents, and use the relation Ji(Pi

0P−i1 )2

to reduce each term to have an exponent of less than 1, introducing extra J terms as needed. Finallyafter reducing the exponents of J and P0, knowing that they are all order n, we arrive at the desiredform.

72

Corollary 5.5.4. |Gn| ≤ n22n−1 for all n ∈N.

Proof. Follows from lemma 5.6.3.

Lemma 5.5.5. |Hn| ≥ n22n−1 for all n ∈N.

Proof. We lower bound the order of the group Hn by exhibiting n22n−1 distinct elements in thegroup. We divide the proof into cases depending on the parity of n.

First note that z2Di ∈ Hn for all i ∈ [n] since

z−4i Ai1A−i

0 Ai+11 A−(i+1)

0 = z−4iz2iD[i]XiX−iz2(i+1)D[i+1]X

i+1X−(i+1) = z2Di,

where in the first equality we use Proposition 5.4.5. This allows us to generate z2Di0 Di1 · · ·Dik−1 if kis odd via

z−4(k−1)/2(z2Di0)(z2Di1) · · · (z

2Dik−1) = z2Di0 Di1 · · ·Dik−1 , (1)

and Di0 Di1 · · ·Dik−1 if k is even by

z−4(k/2)(z2Di0)(z2Di1) · · · (z

2Dik−1) = Di0 Di1 · · ·Dik−1 . (2)

Let n be odd. From (2) we will be able to generate elements of the form z4iDq00 Dq1

1 · · ·Dqn−1n−1 X j

where there are an even number of nonzero qk for i, j ∈ [n]. It should be clear that the elements withi 6= i′ ∈ 0, 1, . . . , (n− 1)/2 will be distinct. For i > (n− 1)/2, we simply note that we can factorout a z2n = −1 and so we get elements of the form z4i′+2Dq0

0 Dq11 · · ·D

qn−1n−1 X j, where there are an odd

number of nonzero qk for i′ ∈ 0, 1, . . . , (n− 3)/2, j ∈ [n]. Each of these will be distinct from eachother as, again, the powers of the nth root of unity will be distinct, and distinct from the previouscase by the parity of the sign matrices. Therefore we are able to lower-bound |Cn| by n22n−1.

If n is even, we will still be able to generate elements of the form z4iDq00 Dq1

1 · · ·Dqn−1n−1 X j where

there are an even number of nonzero qk for i, j ∈ [n]. However, note that for i > (n− 2)/2, we beginto generate duplicates. So from (1) we can generate elements of the form z4i+2Dq0

0 Dq11 · · ·D

qn−1n−1 X j

for i, j ∈ [n] and an odd number of nonzero qk. These will be distinct from the previous elementsby the parity of the sign matrices but again will begin to generate duplicates after i > (n− 2)/2.Therefore we have the lower-bound of n

2 n2n−1 + n2 n2n−1 = n22n−1 elements.

Lemma 5.5.6. There exists a surjective homomorphism f : Gn → Hn.

Proof. Let us define f : J, P0, P1 → Hn by f (J) = z4 I, f (P0) = A0, f (P1) = A1. We show that fcan be extended to a homomorphism from Gn to Hn. Consider the formal extension f of f to thefree group generated by J, P0, P1. We know from the theory of group presentations that f can beextended to a homomorphism if and only if f (r) = I for all relation r in the presentation of Gn.

It is clear that f respects the first five relations of Gn. Now we check the last family of relations:

f (Ji(Pi0P−i

1 )2) = z4i(Ai0A−i

1 )2

= z4i(Xiz−2i(D0X)−i)2

= (XiX−iD[i])2

= D2[i]

= I.

The homomorphism f is surjective because A0, A1 generate the group Hn.

73

Theorem 5.5.7. Hn ∼= Gn for all n ∈N.

Proof. Since f is surjective, then n22n−1 ≤ |Hn| ≤ |Gn| ≤ n22n−1. Thus |Hn| = |Gn|, so thehomomorphism is also injective.

Remark 5.5.8. What about the group generated by Bob’s operators in Sn? We can define

G′n =

⟨Q0, Q1, J | Qn

0 , Qn1 , Jn, [J, Q0], [J, Q1], Ji

(Q−i

0 Q−i1

)2for i = 1, 2, . . . , bn/2c

⟩.

and with a similar argument as in Theorem 5.6.7 show that 〈B0, B1, z4n I〉 ∼= G′n. It is now easily

verified that the mapping P0 7→ Q−10 , P1 7→ Q1, J 7→ J is an isomorphism between Gn and G′n. So

Alice and Bob’s operator generate the same group, that is 〈A0, A1, z4n I〉 = 〈B0, B1, z4

n I〉. The latterfact could also be verified directly.

5.6 Group structure of Sn

Let Hn = 〈A0, A1〉 be the group generated by Alice’s observables in Sn. Note that since (A1A∗0)2 =

z4n I, we could equivalently define Hn = 〈A0, A1, z4

n I〉. Also let

Gn =

⟨P0, P1, J | Pn

0 , Pn1 , Jn, [J, P0], [J, P1], Ji

(Pi

0P−i1

)2for i = 1, 2, . . . , bn/2c

⟩.

In this section we show that Hn ∼= Gn. So it also holds that Hn is a representation of Gn.We conjecture that Gn is a self-test for Gn, in the sense that every optimal strategy of Gn is a|ψ〉-representation of Gn. In Section 5.7, we prove this for n = 3.

Remark 5.6.1. Note that the relations Ji(

Pi0P−i

1

)2holds in Gn for all i.

The following lemma helps us develop a normal form for elements of Gn.

Lemma 5.6.2. For all i, j, the elements Pi0P−i

1 and Pj0P−j

1 commute.

Proof. (Pi

0P−i1

)(Pj

0P−j1

)= J−iPi

1P−i0 Pj

0P−j1

= J−iPi1Pj−i

0 P−j1

= J−iPi1(

Pj−i0 P−(j−i)

1

)P−i

1

= J−i−(j−i)Pi1Pj−i

1 P−(j−i)0 P−i

1

= J−j(Pj1P−j

0

)(Pi

0P−i1

)= J−j(J jPj

0P−j1

)(Pi

0P−i1

)=(

Pj0P−j

1

)(Pi

0P−i1

).

Lemma 5.6.3. For every g ∈ Gn there exist i, j ∈ [n] and qk ∈ 0, 1 for k = 1, 2, . . . , n− 1 such that

g = JiPj0

(P0P−1

1

)q1(

P20 P−2

1

)q2 · · ·(

Pn−10 P−(n−1)

1

)qn−1 .

74

Proof. First note that J is central, therefore we can write g in Gn as

g = JiPj10 Pj2

1 Pj30 · · · P

jk1 ,

for some k ∈N, i ∈ [n], jl ∈ [n] where l = 1, 2, . . . , k. Without loss of generality, let k be even. Weperform the following sequence of manipulations

g = JiPj10 Pj2

1 Pj30 · · · P

jk−21 Pjk−1

0 Pjk1

= JiPj10 Pj2

1 Pj30 · · · P

jk−21 Pjk−1

0 Pjk0

(P−jk

0 Pjk1

)= JiPj1

0 Pj21 Pj3

0 · · · Pjk−21 Pjk−1+jk

1

(P−(jk−1+jk)

1 Pjk−1+jk0

)(P−jk

0 Pjk1

)= Ji−(jk−1+jk)Pj1

0 Pj21 Pj3

0 · · · Pjk−2+jk−1+jk1

(P−(jk−1+jk)

0 Pjk−1+jk1

)(P−jk

0 Pjk1

)= · · ·= Ji−sP−s1

0

(Ps2

0 P−s21

)· · ·(

Psk−10 P−sk−1

1

)(Psk

0 P−sk1

),

where sl = −∑kt=l jt and s = −∑(k−2)/2

t=1 s2t+1. Then we use the commutation relationship fromlemma 5.6.2 to group the terms with the same P0 and P1 exponents, and use the relation Ji(Pi

0P−i1 )2

to reduce each term to have an exponent of less than 1, introducing extra J terms as needed. Finallyafter reducing the exponents of J and P0, knowing that they are all order n, we arrive at the desiredform.

Corollary 5.6.4. |Gn| ≤ n22n−1 for all n ∈N.

Proof. Follows from lemma 5.6.3.

Lemma 5.6.5. |Hn| ≥ n22n−1 for all n ∈N.

Proof. We lower bound the order of the group Hn by exhibiting n22n−1 distinct elements in thegroup. We divide the proof into cases depending on the parity of n.

First note that z2Di ∈ Hn for all i ∈ [n] since

z−4i Ai1A−i

0 Ai+11 A−(i+1)

0 = z−4iz2iD[i]XiX−iz2(i+1)D[i+1]X

i+1X−(i+1) = z2Di,

where in the first equality we use Proposition 5.4.5. This allows us to generate z2Di0 Di1 · · ·Dik−1 if kis odd via

z−4(k−1)/2(z2Di0)(z2Di1) · · · (z

2Dik−1) = z2Di0 Di1 · · ·Dik−1 , (1)

and Di0 Di1 · · ·Dik−1 if k is even by

z−4(k/2)(z2Di0)(z2Di1) · · · (z

2Dik−1) = Di0 Di1 · · ·Dik−1 . (2)

Let n be odd. From (2) we will be able to generate elements of the form z4iDq00 Dq1

1 · · ·Dqn−1n−1 X j

where there are an even number of nonzero qk for i, j ∈ [n]. It should be clear that the elements withi 6= i′ ∈ 0, 1, . . . , (n− 1)/2 will be distinct. For i > (n− 1)/2, we simply note that we can factorout a z2n = −1 and so we get elements of the form z4i′+2Dq0

0 Dq11 · · ·D

qn−1n−1 X j, where there are an odd

number of nonzero qk for i′ ∈ 0, 1, . . . , (n− 3)/2, j ∈ [n]. Each of these will be distinct from eachother as, again, the powers of the nth root of unity will be distinct, and distinct from the previouscase by the parity of the sign matrices. Therefore we are able to lower-bound |Cn| by n22n−1.

If n is even, we will still be able to generate elements of the form z4iDq00 Dq1

1 · · ·Dqn−1n−1 X j where

there are an even number of nonzero qk for i, j ∈ [n]. However, note that for i > (n− 2)/2, we begin

75

to generate duplicates. So from (1) we can generate elements of the form z4i+2Dq00 Dq1

1 · · ·Dqn−1n−1 X j

for i, j ∈ [n] and an odd number of nonzero qk. These will be distinct from the previous elementsby the parity of the sign matrices but again will begin to generate duplicates after i > (n− 2)/2.Therefore we have the lower-bound of n

2 n2n−1 + n2 n2n−1 = n22n−1 elements.

Lemma 5.6.6. There exists a surjective homomorphism f : Gn → Hn.

Proof. Let us define f : J, P0, P1 → Hn by f (J) = z4 I, f (P0) = A0, f (P1) = A1. We show that fcan be extended to a homomorphism from Gn to Hn. Consider the formal extension f of f to thefree group generated by J, P0, P1. We know from the theory of group presentations that f can beextended to a homomorphism if and only if f (r) = I for all relation r in the presentation of Gn.

It is clear that f respects the first five relations of Gn. Now we check the last family of relations:

f (Ji(Pi0P−i

1 )2) = z4i(Ai0A−i

1 )2

= z4i(Xiz−2i(D0X)−i)2

= (XiX−iD[i])2

= D2[i]

= I.

The homomorphism f is surjective because A0, A1 generate the group Hn.

Theorem 5.6.7. Hn ∼= Gn for all n ∈N.

Proof. Since f is surjective, then n22n−1 ≤ |Hn| ≤ |Gn| ≤ n22n−1. Thus |Hn| = |Gn|, so thehomomorphism is also injective.

Remark 5.6.8. What about the group generated by Bob’s operators in Sn? We can define

G′n =

⟨Q0, Q1, J | Qn

0 , Qn1 , Jn, [J, Q0], [J, Q1], Ji

(Q−i

0 Q−i1

)2for i = 1, 2, . . . , bn/2c

⟩.

and with a similar argument as in Theorem 5.6.7 show that 〈B0, B1, z4n I〉 ∼= G′n. It is now easily

verified that the mapping P0 7→ Q−10 , P1 7→ Q1, J 7→ J is an isomorphism between Gn and G′n. So

Alice and Bob’s operator generate the same group, that is 〈A0, A1, z4n I〉 = 〈B0, B1, z4

n I〉. The latterfact could also be verified directly.

5.7 Optimality and rigidity for G3

In this section, we show that S3 is optimal, and therefore ν∗(G3) = 5/6. We also show that G3 isa self-test for the strategy S3. We obtain these results by obtaining algebraic relations betweenoperators in any optimal strategy using an SOS decomposition for B3.

5.7.1 Optimality of S3

For every operator Ai, Bj for which A3i = B3

j = I and [Ai, Bj] = 0, we have the following SOSdecomposition:

6I − A0B∗0 − A∗0 B0 − A0B1 − A∗0 B∗1 − A1B∗0 − A∗1 B0 −ω∗A1B1 −ωA∗1 B∗1= λ1(S∗1S1 + S∗2S2) + λ2(T∗1 T1 + T∗2 T2) + λ3(T∗3 T3 + T∗4 T4) + λ4(T∗5 T5 + T∗6 T6), (5.7.1)

76

where

S1 = A0 + ωA1 + ω∗B0 + ωB∗1 ,S2 = A∗0 + ω∗A∗1 + ωB∗0 + ω∗B1,T1 = A0B∗0 + aiA∗0 B0 − aA0B1 + iA∗0 B∗1 + aA1B∗0 − iA∗1 B0 −ω∗A1B1 − aiωA∗1 B∗1 ,T2 = A0B∗0 + aiA∗0 B0 + aA0B1 − iA∗0 B∗1 − aA1B∗0 + iA∗1 B0 −ω∗A1B1 − aiωA∗1 B∗1 ,T3 = A0B∗0 − aiA∗0 B0 − aA0B1 − iA∗0 B∗1 + aA1B∗0 + iA∗1 B0 −ω∗A1B1 + aiωA∗1 B∗1 ,T4 = A0B∗0 − aiA∗0 B0 + aA0B1 + iA∗0 B∗1 − aA1B∗0 − iA∗1 B0 −ω∗A1B1 + aiωA∗1 B∗1 ,T5 = A0B∗0 + bA∗0 B0 − bA0B1 − A∗0 B∗1 − bA1B∗0 − A∗1 B0 + ω∗A1B1 + bωA∗1 B∗1 ,T6 = 6I − A0B∗0 − A∗0 B0 − A0B1 − A∗0 B∗1 − A1B∗0 − A∗1 B0 −ω∗A1B1 −ωA∗1 B∗1 ,

and

λ1 =5

86, λ2 =

14 +√

214 · 86

, λ3 =14−

√21

4 · 86, λ4 =

786

,

a =2ω + 3ω∗√

7, b =

3ω + 8ω∗

7, ω = ω3.

This SOS decomposition tells us that B3 6I in positive semidefinite order. So from Theorem5.3.1, it holds that ν∗(G3) ≤ 5/6. Combined with Theorem 5.4.9, we have ν∗(G3) = 5/6.

This SOS is obtained from the dual semidefinite program associated with the second level ofthe NPA hierarchy. Surprisingly, the first level of NPA is not enough to obtain this upper bound, aswas the case with CHSH.

5.7.2 Algebraic relations

As in Section ??, we derive group and ring relations for optimal strategies of G3 from the SOS (5.7.1).For the rest of this section, let (A0, A1, B0, B1, |ψ〉) be an optimal strategy. Then 〈ψ|(6I−B3)|ψ〉 = 0.So it also holds that Si|ψ〉 = 0 and Tj|ψ〉 = 0 for all i ∈ [2] and j ∈ [6]. Therefore

(T1 + T2 + T3 + T4)|ψ〉 = 0, (T1 + T2 − T3 − T4)|ψ〉 = 0,(T1 − T2 + T3 − T4)|ψ〉 = 0, (T1 − T2 − T3 + T4)|ψ〉 = 0.

From which by simplification we obtain the four relations

A0B∗0 |ψ〉 = ω∗A1B1|ψ〉, A∗0 B0|ψ〉 = ωA∗1 B∗1 |ψ〉,A0B1|ψ〉 = A1B∗0 |ψ〉, A∗0 B∗1 |ψ〉 = A∗1 B0|ψ〉. (5.7.2)

Now from these four relations and the fact that Ai, Bj are generalized observables satisfying[Ai, Bj] = 0, we obtain

ω∗A∗0 A1|ψ〉 = B∗1 B∗0 |ψ〉 (5.7.3)ωA0A∗1 |ψ〉 = B1B0|ψ〉 (5.7.4)

A∗0 A1|ψ〉 = B0B1|ψ〉 (5.7.5)A0A∗1 |ψ〉 = B∗0 B∗1 |ψ〉 (5.7.6)A∗1 A0|ψ〉 = ω∗B0B1|ψ〉 (5.7.7)A1A∗0 |ψ〉 = ωB∗0 B∗1 |ψ〉 (5.7.8)A∗1 A0|ψ〉 = B∗1 B∗0 |ψ〉 (5.7.9)A1A∗0 |ψ〉 = B1B0|ψ〉. (5.7.10)

77

From the pair of relations (5.7.3) and (5.7.9) as well as the pair of relations (5.7.4) and (5.7.10), weobtain the following relations between Alice’s observables acting on the state |ψ〉:

A∗0 A1|ψ〉 = ωA∗1 A0|ψ〉, (5.7.11)A1A∗0 |ψ〉 = ωA0A∗1 |ψ〉. (5.7.12)

Next we prove two propositions regarding H = H3 = ωA0A1A0 + ωA∗0 A1 + ωA1A∗0 definedin (??).

Proposition 5.7.1. (H + H∗)|ψ〉 = −2|ψ〉Proof. We start by writing

(ωB∗0 + ω∗B1 + B0B∗1 + B∗1 B0)|ψ〉 = (ω∗B0 + ωB∗1)(ω∗B0 + ωB∗1)|ψ〉

= −(ω∗B0 + ωB∗1)(A0 + ωA1)|ψ〉= −(A0 + ωA1)(ω

∗B0 + ωB∗1)|ψ〉= (A0 + ωA1)(A0 + ωA1)|ψ〉= (A∗0 + ω∗A∗1 + ωA0A1 + ωA1A0)|ψ〉,

where for the second and fourth equality, we used the relation S1|ψ〉 = 0, and for the third equalitywe used the fact that Alice and Bob’s operators commute. Now using S2|ψ〉 = 0, we obtain

(B0B∗1 + B∗1 B0)|ψ〉 = (2A∗0 + 2ω∗A∗1 + ωA0A1 + ωA1A0)|ψ〉. (5.7.13)

Similarly we have

(B1B∗0 + B∗0 B1)|ψ〉 = (2A0 + 2ωA1 + ω∗A∗0 A∗1 + ω∗A∗1 A∗0)|ψ〉. (5.7.14)

We proceed by simplifying T6|ψ〉 = 0 using relations (5.7.2) to obtain

(3I − A0B∗0 − A∗0 B0 − A0B1 − A∗0 B∗1)|ψ〉 = 0.

Let P = A0B∗0 + A∗0 B0 + A0B1 + A∗0 B∗1 , and write

0 =(3I − A0B∗0 − A∗0 B0 − A0B1 − A∗0 B∗1

)∗(3I − A0B∗0 − A∗0 B0 − A0B1 − A∗0 B∗1)|ψ〉

=(13I − 5P + A∗0(B1B∗0 + B∗0 B1) + A0(B0B∗1 + B∗1 B0) + B∗0 B∗1 + B0B1 + B1B0 + B∗1 B∗0

)|ψ〉

= (−2I + A∗0(B1B∗0 + B∗0 B1) + A0(B0B∗1 + B∗1 B0) + B∗0 B∗1 + B0B1 + B1B0 + B∗1 B∗0)|ψ〉, (5.7.15)

where in the last line, we used (3I − P)|ψ〉 = 0. Using identities (5.7.13) and (5.7.14)(A∗0(B1B∗0 + B∗0 B1) + A0(B0B∗1 + B∗1 B0)

)|ψ〉

=(4I + ωA0A1A0 + ω∗A∗0 A∗1 A∗0 + 2ωA∗0 A1 + ω∗A0A∗1 + 2ω∗A0A∗1 + ωA∗0 A1

)|ψ〉.

Transferring Bob’s operators to Alice using identities (5.7.3-5.7.6)(B∗0 B∗1 + B0B1 + B1B0 + B∗1 B∗0

)|ψ〉 =

(A0A∗1 + A∗0 A1 + ωA0A∗1 + ω∗A∗0 A1

)|ψ〉.

Plugging these back in (5.7.15)

0 = (2I + ωA0A1A0 + ω∗A∗0 A∗1 A∗0 + (3ω + ω∗ + 1)A∗0 A1 + (3ω∗ + ω + 1)A0A∗1)|ψ〉= (2I + ωA0A1A0 + ω∗A∗0 A∗1 A∗0 + 2ωA∗0 A1 + 2ω∗A0A∗1)|ψ〉= (2I + ωA0A1A0 + ω∗A∗0 A∗1 A∗0 + ωA∗0 A1 + ω∗A∗1 A0 + ω∗A0A∗1 + ωA1A∗0)|ψ〉.= (2I + H + H∗)|ψ〉,

where in the first line we used 1 + ω + ω∗ = 0, and in the second line we used identities (5.7.11)and (5.7.12).

78

Proposition 5.7.2. (H + I)|ψ〉 = (H∗ + I)|ψ〉 = 0.

Proof. First note

〈ψ|H∗H|ψ〉 = 〈ψ|(3I + A∗0 A∗1 A0A1 + A∗1 A∗0 A1A0 + A∗1 A0A1A∗0 + A0A∗1 A∗0 A1

+ A∗0 A∗1 A∗0 A1A∗0 + A0A∗1 A0A1A0)|ψ〉. (5.7.16)

Using (5.7.11) and (5.7.12), we have

〈ψ|A0A∗1 A∗0 A1|ψ〉 = ω〈ψ|A0A∗1 A∗1 A0|ψ〉 = ω〈ψ|A0A1A0|ψ〉,〈ψ|A∗0 A∗1 A∗0 A1A∗0 |ψ〉 = ω〈ψ|A∗0 A∗1 A∗0 A0A∗1 |ψ〉 = ω〈ψ|A∗0 A1|ψ〉,

and using (5.7.5) and (5.7.7)

〈ψ|A∗0 A∗1 A0A1|ψ〉 = 〈ψ|A∗0 A1A1A∗0 A∗0 A1|ψ〉 = ω〈ψ|B∗1 B∗0 A1A∗0 B0B1|ψ〉 = ω〈ψ|A1A∗0 |ψ〉,

and taking conjugate transpose of these three we obtain

〈ψ|A∗1 A0A1A∗0 |ψ〉 = ω∗〈ψ|A∗0 A∗1 A∗0 |ψ〉,〈ψ|A0A∗1 A0A1A0|ψ〉 = ω∗〈ψ|A∗1 A0|ψ〉,〈ψ|A∗1 A∗0 A1A0|ψ〉 = ω∗〈ψ|A0A∗1 |ψ〉.

Plugging these back in (5.7.16), we obtain

‖H|ψ〉‖2 = 〈ψ|H∗H|ψ〉= 〈ψ|(3I + ωA0A1A0 + ωA∗0 A1 + ωA1A∗0 + ω∗A∗0 A∗1 A∗0 + ω∗A∗1 A0 + ω∗A0A∗1)|ψ〉= 〈ψ|(3I + H + H∗)|ψ〉= 〈ψ|I|ψ〉= 1,

where in fourth equality we used Proposition 5.7.1. Similarly ‖H∗|ψ〉‖ = 1. From (H + H∗)|ψ〉 =−2|ψ〉 and the fact that H|ψ〉 and H∗|ψ〉 are unit vectors, we get that H|ψ〉 = H∗|ψ〉 = −|ψ〉.

Proposition 5.7.3. A0A1A0|ψ〉 = ωA∗0 A∗1 A∗0 |ψ〉.

Proof. By Proposition 5.7.2, H|ψ〉 = H∗|ψ〉, and by identities (5.7.11), (5.7.12), (ωA∗0 A1 +ωA1A∗0)|ψ〉 =(ω∗A∗1 A0 + ω∗A0A∗1)|ψ〉. Putting these together, we obtain A0A1A0|ψ〉 = ωA∗0 A∗1 A∗0 |ψ〉.

Proposition 5.7.4. A0A∗1 A∗0 A1|ψ〉 = A∗0 A1A0A∗1 |ψ〉 in other words A0A∗1 and A∗0 A1 commute on |ψ〉

Proof. To see this write

A0A∗1 A∗0 A1|ψ〉 = ωA0A∗1 A∗1 A0|ψ〉= ωA0A1A0|ψ〉= ω∗A∗0 A∗1 A∗0 |ψ〉= ω∗A∗0 A1A1A∗0 |ψ〉= A∗0 A1A0A∗1 |ψ〉,

where in the first line we used 5.7.11, in the third line we used 5.7.3, and in the fifth line we used5.7.12.

79

5.7.3 Rigidity of G3

Suppose (A0, A1, B0, B1, |ψ〉) is an optimal strategy for G3. By Theorem 5.6.7, we know thatthe optimal operators of Alice defined in section 5.4.1 generate the group

G3 =⟨

J, P0, P1 : J3, P30 , P3

1 , [J, P0], [J, P1], J(P0P−11 )2

⟩,

The same group is generated by Bob’s operators as in Remark 5.6.8. We apply Corollary 5.2.5 withGA = GB = G3. In order to do this, we first prove the following lemma stating that every optimalstrategy is a |ψ〉-representation of G.

Lemma 5.7.5. Let (A0, A1, B0, B1, |ψ〉) be an optimal strategy for G3. Define maps fA, fB : G3 →Ud(C) by

fA(J) = ω3 I, fA(P0) = A0, fA(P0P−11 ) = A0A∗1 , fA(P−1

0 P1) = A∗0 A1

fB(J) = ω3 I, fB(P0) = B∗0 , fB(P0P−11 ) = B∗0 B∗1 , fB(P−1

0 P1) = B0B1

and extend it to all of G3 using the normal form from Lemma 5.6.3. Then fA, fB are |ψ〉-representations ofG3.

Proof. These maps are well defined since every element of G3 can be written uniquly as

JiPj0

(P0P−1

1

)q1(

P−10 P1

)q2

for i, j ∈ [3], q1, q2 ∈ [2]. All we need is that fA(g) fA(g′)|ψ〉 = fA(gg′)|ψ〉 for all g, g′ ∈ G3. Theproof is reminiscent of the proof that gg′ can be written in normal form for every g, g′ ∈ G3. Exceptthat we need to be more careful here, since we are dealing with Alice’s operators A0, A1, and notthe abstract group elements P0, P1. Therefore we can only use the state-dependent relations derivedin the previous section. We must show that

fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2) fA(Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)|ψ〉

= fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2 Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)|ψ〉 (5.7.17)

for all i, j, i′, j′ ∈ [3] and q1, q2, q′1, q′2 ∈ [2].

Claim 9. Without loss of generality, we can assume i = j = i′ = q′1 = q′2 = 0.

Proof. Fix i, j, q1, q2, i′, j′, q′1, q′2. We first show that without loss of generality we can assume q′1 =q′2 = 0. By Lemma 5.6.3, there exist i′′, j′′ ∈ [3], q′′1 , q′′2 ∈ [2] such that(

JiPj0(P0P−1

1 )q1(P−10 P1)

q2)(

Ji′Pj′0

)= Ji′′Pj′′

0 (P0P−11 )q′′1 (P−1

0 P1)q′′2 .

So it also holds that(JiPj

0(P0P−11 )q1(P−1

0 P1)q2)(

Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)= Ji′′Pj′′

0 (P0P−11 )q′′1+q′1(P−1

0 P1)q′′2+q′2

since by Lemma 5.6.2, P0P−11 and P−1

0 P1 commute. So the right-hand-side of (5.7.17) can be written

fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2 Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)|ψ〉

= fA(Ji′′Pj′′0 (P0P−1

1 )q′′1+q′1(P−10 P1)

q′′2+q′2)|ψ〉

= ωi′′Aj′′0 (A0A−1

1 )q′′1+q′1(A−10 A1)

q′′2+q′2 |ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 ωi′′Aj′′0 (A0A−1

1 )q′′1 (A−10 A1)

q′′2 |ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 fA(Ji′′Pj′′0 (P0P−1

1 )q′′1 (P−10 P1)

q′′2 )|ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 fA((JiPj0(P0P−1

1 )q1(P−10 P1)

q2)(Ji′Pj′0 ))|ψ〉,

80

where in the fourth equality, we used (5.7.5) and (5.7.6) and the fact that Alice and Bob’s operatorscommute.

Also since Alice and Bob’s operators commute

fA(Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)|ψ〉 = ωi′Aj′0 (A0A∗1)

q′1(A∗0 A1)q′2 |ψ〉

= (B0B1)q′2 ωi′Aj′

0 (A0A∗1)q′1 |ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 ωi′Aj′0 |ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 fA(Ji′Pj′0 )|ψ〉.

Therefore the left-hand-side of (5.7.17) can be written as

fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2) fA(Ji′Pj′0 (P0P−1

1 )q′1(P−10 P1)

q′2)|ψ〉

= (B0B1)q′2(B∗0 B∗1)

q′1 fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2) fA(Ji′Pj′0 )|ψ〉

Since B0, B1 are unitaries, (5.7.17) is equivalent to the following identity

fA(JiPj0(P0P−1

1 )q1(P−10 P1)

q2) fA(Ji′Pj′0 )|ψ〉 = fA((JiPj

0(P0P−11 )q1(P−1

0 P1)q2)(Ji′Pj′

0 ))|ψ〉,

in other words we can assume without loss of generality q′1 = q′2 = 0. The case of i = j = 0 ishandled similarly. Also since J and f (J) are both central, we can assume i′ = 0.

By this claim, we just need to verify

fA((P0P−11 )q1(P−1

0 P1)q2) fA(Pj′

0 )|ψ〉 = fA((P0P−11 )q1(P−1

0 P1)q2 Pj′

0 )|ψ〉 (5.7.18)

There are 12 cases to consider: q1, q2 ∈ [2], j′ ∈ [3]. The case of j′ = 0 is trivial, and the case ofj′ = 2 is handled similar to the case of j′ = 1. So we only consider the case of j′ = 1. The case ofq1 = q2 = 0 is trivial. We analyse the remaining three cases one-by-one:

• q1 = 0, q2 = 1: First note that

(P−10 P1)P0 = P0P0P−1

1 P−11 P0 = J2P0(P0P−1

1 )(P−10 P1),

which allows us to write

fA((P−10 P1)) fA(P0)|ψ〉 = A∗0 A1A0|ψ〉

= A∗0 A∗1 A∗1 A0|ψ〉= ω∗A∗0 A∗1 A∗0 A1|ψ〉= ω∗A0(A0A∗1)(A∗0 A1)|ψ〉= fA(J2P0(P0P−1

1 )(P−10 P1))|ψ〉

= fA((P−10 P1)P0)|ψ〉,

where in the third line we used (5.7.11).

• q1 = 1, q2 = 0:

(P0P−11 )P0 = J2P0(P−1

0 P1)

81

which allows us to write

fA(P0P−11 ) fA(P0)|ψ〉 = (A0A∗1)A0|ψ〉

= A0(A∗1 A0)|ψ〉= ω∗A0(A∗0 A1)|ψ〉= fA(J2P0(P−1

0 P1))|ψ〉= fA((P0P−1

1 )P0)|ψ〉,

where in the third line we used (5.7.11).

• q1 = q2 = 1:

(P0P−11 )(P−1

0 P1)P0 = J(P0P−11 )(P−1

1 P0)P0 = JP0(P1P−10 ) = J2P0(P0P−1

1 ).

Now write

fA((P0P−11 )(P−1

0 P1)) fA(P0)|ψ〉 = A0A∗1 A∗0 A1A0|ψ〉= A0A∗1 A0A0A1A0|ψ〉= ωA0A∗1 A0A∗0 A∗1 A∗0 |ψ〉= ωA0(A1A∗0)|ψ〉= ω∗A0(A0A∗1)|ψ〉= fA(J2P0(P0P−1

1 ))|ψ〉= fA((P0P−1

1 )(P−10 P1)P0)|ψ〉,

where in the third line we used Proposition 5.7.3 and in the second last line we used (5.7.12).

The proof that fB is a |ψ〉-representation follows similarly.

Theorem 5.7.6. G3 is rigid.

Proof. The representation theory of G3 is simple. There are nine irreducible representation ofdimension one: These are given by P0 7→ ωi, P1 7→ ω j, J 7→ ω2(j−i) for i, j ∈ [3]. It also has threeirreducible representations g1, g2, g3 of dimension three defined by

g1(P0) =

0 0 11 0 00 1 0

, g1(P1) =

0 0 ω∗

−ω∗ 0 00 −ω∗ 0

, g1(J) =

ω 0 00 ω 00 0 ω

,

g2(P0) =

0 0 11 0 00 1 0

, g2(P1) =

0 0 −1−1 0 00 1 0

, g2(J) =

1 0 00 1 00 0 1

,

g3(P0) =

0 1 00 0 11 0 0

, g3(P1) =

0 ω 00 0 −ω−ω 0 0

, g3(J) =

ω∗ 0 00 ω∗ 00 0 ω∗

.

Among these g1, is the only representation that gives rise to an optimal strategy. This followsfrom a simple enumeration of these 12 irreducible representations. However we could alsoimmediately see this, since g1 is the only irreducible representation that satisfies the ring relationH3 + I = 0.

82

Define a unitarily equivalent irreducible representation g′1 = Ug1U∗ where U =

0 1 01 0 00 0 1

.

Now A0 = g1(P0), A1 = g1(P1), B0 = g′1(P0)∗, B1 := g′1(P1) is the same strategy defined in example5.4.2.

In addition|ψ3〉 =

1√10

((1− z4)|00〉+ 2|12〉+ (1 + z2)|21〉

)is the unique state that maximizes ν(G3,Sg1,g′1,|ψ〉). This follows since |ψ3〉 is the unique eigenvec-tor associated with the largest eigenvalue of B3(A0, A1, B0, B1). The rigidity of G3 follows fromCorollary 5.2.5.

Remark 5.7.7. The game G3 is in fact a robust self-test. We omit the proof, but at a high-level, if astrategy (A0, A1, B0, B1, |ψ〉) is ε-optimal for G3, then

〈ψ|(6I −B3)|ψ〉 ≤ O(ε).

Consequently, ‖Si|ψ〉‖ ≤ O(√

ε), ‖Tj|ψ〉‖ ≤ O(√

ε) for all i ∈ [2], j ∈ [6]. From which one obtains arobust version of every relation in this section.

5.8 SOS approach to solution group

In this section we show that the connection between an LCS game over Z2 and its solution groupshown in [CLS17] can be determined using sum of squares techniques.

We will suppress the tensor product notation and simply represent a strategy for an LCS gameGA,b by a state |ψ〉 ∈ H and a collection of commuting measurement systems Ei,x and Fj,y.Using the notation outlined in section 5.2.3 we define the following sets of observables

• Alice’s Observables: A(i)j = ∑x:xj=1 Ei,x −∑x:xj=−1 Ei,x, for each i ∈ [r] and j ∈ Vi

• Bob’s Observables: Bj = Fj,1 − Fj,−1 for each j ∈ [s].

Note A(i)j commutes with A(i)

j′ for all i ∈ [r] and j, j′ ∈ Vi and Bj commutes with A(i)j for all i, j.

These observables will satisfy the following identities:

∑x:x∈Si

Ei,x =12

(I + (−1)bi ∏

k∈Vi

A(i)k

)(5.8.1)

∑x:y=xj

Ei,x =12

(I + yA(i)

j

)(5.8.2)

83

The probability of Alice and Bob winning the game is given by evaluating 〈ψ|v|ψ〉 where

v = ∑i∈[r]j∈Vi

1r|Vi|

∑x,y:

x∈Siy=xj

Ei,xFj,y

= ∑i,j

12r|Vi|

1− ∑x,y:

x∈Siy=xj

Ei,xFj,y

2

.

Observe using identities 5.8.1 and 5.8.2 we have

1− ∑x,y:

x∈Siy=xj

Ei,xFj,y

= I −∑y

Fj,y ∑x:

x∈Siy=xj

Ei,x

= I − 14 ∑

yFj,y

((I + yA(i)

j )(I + (−1)bi ∏k∈vi

A(i)k )

)

= I − 14 ∑

yFj,y

(I + yA(i)

j + (−1)bi ∏k∈vi

A(i)k + y(−1)bi ∏

k∈vi

A(i)k A(i)

j

)

= I − 14

Fj,1

(I + A(i)

j + (−1)bi ∏k∈vi

A(i)k + (−1)bi ∏

k∈vi

A(i)k A(i)

j

)

− 14

Fj,−1

(I − A(i)

j + (−1)bi ∏k∈vi

A(i)k +−(−1)bi ∏

k∈vi

A(i)k A(i)

j

)

= I − 14

I − 14

Bj A(i)j −

14(−1)bi ∏

k∈vi

A(i)k −

14

Bj(−1)bi ∏k∈vi

A(i)k A(i)

j

=18

((I − Bj A

(i)j )2 + (I − (−1)bi ∏

k∈vi

A(i)k )2 + (I − (−1)bi ∏

k∈vi

A(i)k A(i)

j Bj)2

).

Thus Alice and Bob are using a perfect strategy if and only if

0 = (I − Bj A(i)j )|ψ〉 = (I − (−1)bi ∏

k∈vi

A(i)k )|ψ〉 = (I − (−1)bi ∏

k∈vi

A(i)k A(i)

j Bj)|ψ〉.

The above equalities will hold exactly when the following two identities hold for all i and j ∈ Vi,

Bj|ψ〉 = A(i)j |ψ〉 (5.8.3)

|ψ〉 = (−1)bi ∏k∈vi

A(i)k |ψ〉 (5.8.4)

Using identities 5.8.3 and 5.8.4 it is possible to define a |ψ〉-representation for the solution groupGA,b.

84

5.9 A non-rigid pseudo-telepathic LCS game

The canonical example of a pseudo-telepathic LCS games is the Mermin-Peres magic square game[Mer90] defined in the following figure.

e1 — e2 — e3| | ||

e4 — e5 — e6| | ||

e7 — e8 — e9

Figure 5.3: This describes the Mermin-Peres magic square game. Each single-line indicates that thevariables along the line multiply to 1, and the double-line indicates that the variables along the linemultiply to −1.

It is well-known that the Mermin-Peres magic square game has the following operator solutionfor which the corresponding quantum strategy is rigid [WBMS15].

A1 = I ⊗ σZ, A2 = σZ ⊗ I, A3 = σZ ⊗ σZ

A4 = σX ⊗ I, A5 = I ⊗ σX, A6 = σX ⊗ σX

A7 = σX ⊗ σZ, A8 = σZ ⊗ σX, A9 = σY ⊗ σY,

In this section, we provide an example of a non-local game whose perfect solutions must obeyparticular group relations but is not a self-test. This game, glued magic square, is described in Figure5.4.

e1 — e2 — e3| | ||

e4 — e5 — e6| | ||

e7 — e8 — e9||

e10 — e11 — e12|| | |

e13 — e14 — e15|| | |

e16 — e17 — e18

Figure 5.4: This describes a LCS game with 18 variables e1, e2, . . . , e18. Each single-line indicatesthat the variables along the line multiply to 1, and the double-line indicates that the variables alongthe line multiply to −1.

In order to show that this game is not a self-test, we first define two operator solutions, that

85

give rise to perfect strategies. Let E = E1, E2, . . . , E18 be defined as

Ei =

(I4 00 Ai

)for i = 1, 2, . . . , 9(

Ai−9 00 I4

)for i = 10, 11, . . . , 18

and F = F1, F2, . . . , F18 as

Fi =

Ai for i = 1, 2 . . . , 9I4 for i = 10, 11 . . . , 18

These two operators solutions E and F give rise to two quantum strategies with the entangledstates |ψ1〉 = 1√

8 ∑7i=0|i〉|i〉 and |ψ2〉 = 1

2 ∑3i=0|i〉|i〉.

Theorem 5.9.1. The glued magic square game is not a self-test for any quantum strategy.

Proof. Suppose, for the sake of contradiction, there is a quantum strategy(Aii, Bjj|ψ〉

)that is

rigid. Then there exist local isometries UA, UB and VA, VB such that

(UAE1 ⊗UB)|ψ1〉 = ((A1 ⊗ I)|ψ〉)|junk1〉 (5.9.1)(UAE5 ⊗UB)|ψ1〉 = ((A5 ⊗ I)|ψ〉)|junk1〉 (5.9.2)(VAF1 ⊗VB)|ψ2〉 = ((A1 ⊗ I)|ψ〉)|junk2〉 (5.9.3)(VAF5 ⊗VB)|ψ2〉 = ((A5 ⊗ I)|ψ〉)|junk2〉. (5.9.4)

From relation (5.9.2), we obtain

〈ψ1|(E5U∗A ⊗U∗B) = 〈junk1|(〈ψ|(A∗5 ⊗ I)),

and hence together with relation (5.9.1), we obtain the following relation between E5E1 and A∗5 A1

〈ψ1|(E5E1 ⊗ I)|ψ1〉 = 〈ψ|(A∗5 A1 ⊗ I)|ψ〉.Similarly, we also obtain

〈ψ2|(F5F1 ⊗ I)|ψ2〉 = 〈ψ|(A∗5 A1 ⊗ I)|ψ〉,and hence

〈ψ1|(E5E1 ⊗ I)|ψ1〉 = 〈ψ2|(F5F1 ⊗ I)|ψ2〉.By first applying the adjoint to relation (5.9.1) and (5.9.3), we obtain

〈ψ1|(E1E5 ⊗ I)|ψ1〉 = 〈ψ2|(F1F5 ⊗ I)|ψ2〉.Now, since F1 and F5 anti-commute, we get the following relation between E5E1 and E1E5

〈ψ1|(E5E1 ⊗ I)|ψ1〉 = −〈ψ1|(E1E5 ⊗ I)|ψ1〉.However, a direct computation of 〈ψ1|(E5E1 ⊗ I)|ψ1〉 shows that

〈ψ1|(E5E1 ⊗ I)|ψ1〉 =18

7

∑i=0〈i|E5E1|i〉 =

18

Tr(E5E1) =18

Tr(E1E5) = 〈ψ1|(E1E5 ⊗ I)|ψ1〉,

and Tr(E1E5) = Tr(I4) + Tr(I ⊗ σZσX) = 4 6= 0. Hence, the glued magic square game is notrigid.

Although this game is not a self-test, we know from Section 5.8 Alice’s operators must providea |ψ〉-representation for the solution group of glued magic square, and thus must satisfy particulargroup relations.

86

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